symmetric spaces and cartan's classification...the killing form decomposition of symmetric...
TRANSCRIPT
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Symmetric spaces and Cartan’s classification
Henry Twiss
University of Minnesota
April 2020
1 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Outline
1 History
2 Symmetric Spaces
3 Examples of Symmetric Spaces
4 Curvature and Locally Symmetric Spaces
5 Effective Orthogonal Symmetric Lie Algebras and the KillingForm
6 Decomposition of Symmetric Spaces
7 Rank and Classification
8 References
2 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Outline
1 History
2 Symmetric Spaces
3 Examples of Symmetric Spaces
4 Curvature and Locally Symmetric Spaces
5 Effective Orthogonal Symmetric Lie Algebras and the KillingForm
6 Decomposition of Symmetric Spaces
7 Rank and Classification
8 References
3 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
History
Riemann showed that locally there is only one constantcurvature geometry. The most natural geometries to studynext are symmetric spaces. Elie Cartan alone classifiedsymmetric spaces.In the following we will introduce symmetric spaces, give afew prototypical examples, and discuss Cartan’s classification.We will assume throughout that every Lie algebra is a real Liealgebra unless otherwise specified.
4 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
History
Riemann showed that locally there is only one constantcurvature geometry. The most natural geometries to studynext are symmetric spaces. Elie Cartan alone classifiedsymmetric spaces.In the following we will introduce symmetric spaces, give afew prototypical examples, and discuss Cartan’s classification.We will assume throughout that every Lie algebra is a real Liealgebra unless otherwise specified.
4 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
History
Riemann showed that locally there is only one constantcurvature geometry. The most natural geometries to studynext are symmetric spaces. Elie Cartan alone classifiedsymmetric spaces.In the following we will introduce symmetric spaces, give afew prototypical examples, and discuss Cartan’s classification.We will assume throughout that every Lie algebra is a real Liealgebra unless otherwise specified.
4 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
History
Riemann showed that locally there is only one constantcurvature geometry. The most natural geometries to studynext are symmetric spaces. Elie Cartan alone classifiedsymmetric spaces.In the following we will introduce symmetric spaces, give afew prototypical examples, and discuss Cartan’s classification.We will assume throughout that every Lie algebra is a real Liealgebra unless otherwise specified.
4 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
History
Riemann showed that locally there is only one constantcurvature geometry. The most natural geometries to studynext are symmetric spaces. Elie Cartan alone classifiedsymmetric spaces.In the following we will introduce symmetric spaces, give afew prototypical examples, and discuss Cartan’s classification.We will assume throughout that every Lie algebra is a real Liealgebra unless otherwise specified.
4 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Outline
1 History
2 Symmetric Spaces
3 Examples of Symmetric Spaces
4 Curvature and Locally Symmetric Spaces
5 Effective Orthogonal Symmetric Lie Algebras and the KillingForm
6 Decomposition of Symmetric Spaces
7 Rank and Classification
8 References
5 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Introducing Symmetric Spaces
Definition 2.1
A Riemannian manifold M is a symmetric space if for eachp ∈ M, there exists an isometry sp ∈ Iso(M)p such that
s∗,p : TpM → TpM
is the negative of the identity map. The map sp is called asymmetry at p.
Geodesics are preserved by isometries, so sp ◦ γ is a geodesicfor all geodesics γ. Since
(sp ◦ γ)(t) = γ(−t),
symmetric spaces as spaces where at any point there exists asymmetry reversing geodesics through that point. Thisobservation tells us more.
6 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Introducing Symmetric Spaces
Definition 2.1
A Riemannian manifold M is a symmetric space if for eachp ∈ M, there exists an isometry sp ∈ Iso(M)p such that
s∗,p : TpM → TpM
is the negative of the identity map. The map sp is called asymmetry at p.
Geodesics are preserved by isometries, so sp ◦ γ is a geodesicfor all geodesics γ. Since
(sp ◦ γ)(t) = γ(−t),
symmetric spaces as spaces where at any point there exists asymmetry reversing geodesics through that point. Thisobservation tells us more.
6 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Introducing Symmetric Spaces
Definition 2.1
A Riemannian manifold M is a symmetric space if for eachp ∈ M, there exists an isometry sp ∈ Iso(M)p such that
s∗,p : TpM → TpM
is the negative of the identity map. The map sp is called asymmetry at p.
Geodesics are preserved by isometries, so sp ◦ γ is a geodesicfor all geodesics γ. Since
(sp ◦ γ)(t) = γ(−t),
symmetric spaces as spaces where at any point there exists asymmetry reversing geodesics through that point. Thisobservation tells us more.
6 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Introducing Symmetric Spaces
Definition 2.1
A Riemannian manifold M is a symmetric space if for eachp ∈ M, there exists an isometry sp ∈ Iso(M)p such that
s∗,p : TpM → TpM
is the negative of the identity map. The map sp is called asymmetry at p.
Geodesics are preserved by isometries, so sp ◦ γ is a geodesicfor all geodesics γ. Since
(sp ◦ γ)(t) = γ(−t),
symmetric spaces as spaces where at any point there exists asymmetry reversing geodesics through that point. Thisobservation tells us more.
6 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Introducing Symmetric Spaces
Definition 2.1
A Riemannian manifold M is a symmetric space if for eachp ∈ M, there exists an isometry sp ∈ Iso(M)p such that
s∗,p : TpM → TpM
is the negative of the identity map. The map sp is called asymmetry at p.
Geodesics are preserved by isometries, so sp ◦ γ is a geodesicfor all geodesics γ. Since
(sp ◦ γ)(t) = γ(−t),
symmetric spaces as spaces where at any point there exists asymmetry reversing geodesics through that point. Thisobservation tells us more.
6 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Homogeneous Description
1 M is geodesically complete: domains of geodesicsγ : [0, s) are extended by reflecting using symmetries sγ(t)
for t ∈ (s/2, s).
2 Iso(M)◦ acts transitively on M: connect p and q by ageodesic. Letting m be the midpoint of this geodesic,sm(p) = q.
Definition 2.2
A Riemannian manifold M is a homogeneous space if Iso(M)◦
acts transitively on M.
In fact, the second property can be strengthened.
7 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Homogeneous Description
1 M is geodesically complete: domains of geodesicsγ : [0, s) are extended by reflecting using symmetries sγ(t)
for t ∈ (s/2, s).
2 Iso(M)◦ acts transitively on M: connect p and q by ageodesic. Letting m be the midpoint of this geodesic,sm(p) = q.
Definition 2.2
A Riemannian manifold M is a homogeneous space if Iso(M)◦
acts transitively on M.
In fact, the second property can be strengthened.
7 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Homogeneous Description
1 M is geodesically complete: domains of geodesicsγ : [0, s) are extended by reflecting using symmetries sγ(t)
for t ∈ (s/2, s).
2 Iso(M)◦ acts transitively on M: connect p and q by ageodesic. Letting m be the midpoint of this geodesic,sm(p) = q.
Definition 2.2
A Riemannian manifold M is a homogeneous space if Iso(M)◦
acts transitively on M.
In fact, the second property can be strengthened.
7 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Homogeneous Description
1 M is geodesically complete: domains of geodesicsγ : [0, s) are extended by reflecting using symmetries sγ(t)
for t ∈ (s/2, s).
2 Iso(M)◦ acts transitively on M: connect p and q by ageodesic. Letting m be the midpoint of this geodesic,sm(p) = q.
Definition 2.2
A Riemannian manifold M is a homogeneous space if Iso(M)◦
acts transitively on M.
In fact, the second property can be strengthened.
7 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Homogeneous Description
1 M is geodesically complete: domains of geodesicsγ : [0, s) are extended by reflecting using symmetries sγ(t)
for t ∈ (s/2, s).
2 Iso(M)◦ acts transitively on M: connect p and q by ageodesic. Letting m be the midpoint of this geodesic,sm(p) = q.
Definition 2.2
A Riemannian manifold M is a homogeneous space if Iso(M)◦
acts transitively on M.
In fact, the second property can be strengthened.
7 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Homogeneous Description
1 M is geodesically complete: domains of geodesicsγ : [0, s) are extended by reflecting using symmetries sγ(t)
for t ∈ (s/2, s).
2 Iso(M)◦ acts transitively on M: connect p and q by ageodesic. Letting m be the midpoint of this geodesic,sm(p) = q.
Definition 2.2
A Riemannian manifold M is a homogeneous space if Iso(M)◦
acts transitively on M.
In fact, the second property can be strengthened.
7 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Homogeneous Description
1 M is geodesically complete: domains of geodesicsγ : [0, s) are extended by reflecting using symmetries sγ(t)
for t ∈ (s/2, s).
2 Iso(M)◦ acts transitively on M: connect p and q by ageodesic. Letting m be the midpoint of this geodesic,sm(p) = q.
Definition 2.2
A Riemannian manifold M is a homogeneous space if Iso(M)◦
acts transitively on M.
In fact, the second property can be strengthened.
7 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Homogeneous Description
Theorem 2.1
A symmetric space M is precisely a homogeneous space with asymmetry sp at some p ∈ M.
Proof. We are left to show that a homogeneous space with asymmetry is symmetric. Let g ∈ Iso(M)◦ be an isometrytaking p to q. By the chain rule,
sq := g ◦ sp ◦ g−1
defines a symmetry at q.
8 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Homogeneous Description
Theorem 2.1
A symmetric space M is precisely a homogeneous space with asymmetry sp at some p ∈ M.
Proof. We are left to show that a homogeneous space with asymmetry is symmetric. Let g ∈ Iso(M)◦ be an isometrytaking p to q. By the chain rule,
sq := g ◦ sp ◦ g−1
defines a symmetry at q.
8 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Homogeneous Description
Theorem 2.1
A symmetric space M is precisely a homogeneous space with asymmetry sp at some p ∈ M.
Proof. We are left to show that a homogeneous space with asymmetry is symmetric. Let g ∈ Iso(M)◦ be an isometrytaking p to q. By the chain rule,
sq := g ◦ sp ◦ g−1
defines a symmetry at q.
8 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Homogeneous Description
Theorem 2.1
A symmetric space M is precisely a homogeneous space with asymmetry sp at some p ∈ M.
Proof. We are left to show that a homogeneous space with asymmetry is symmetric. Let g ∈ Iso(M)◦ be an isometrytaking p to q. By the chain rule,
sq := g ◦ sp ◦ g−1
defines a symmetry at q.
8 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Isotropy Description
Theorem 2.2
Fixing a basepoint p ∈ M,
M ∼= Iso(M)◦/Iso(M)p.
Iso(M)◦/Iso(M)p is not necessarily a Lie group despiteIso(M)◦ being a connected Lie group. Indeed, Iso(M)p neednot be a normal subgroup.
9 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Isotropy Description
Theorem 2.2
Fixing a basepoint p ∈ M,
M ∼= Iso(M)◦/Iso(M)p.
Iso(M)◦/Iso(M)p is not necessarily a Lie group despiteIso(M)◦ being a connected Lie group. Indeed, Iso(M)p neednot be a normal subgroup.
9 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Isotropy Description
Theorem 2.2
Fixing a basepoint p ∈ M,
M ∼= Iso(M)◦/Iso(M)p.
Iso(M)◦/Iso(M)p is not necessarily a Lie group despiteIso(M)◦ being a connected Lie group. Indeed, Iso(M)p neednot be a normal subgroup.
9 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Outline
1 History
2 Symmetric Spaces
3 Examples of Symmetric Spaces
4 Curvature and Locally Symmetric Spaces
5 Effective Orthogonal Symmetric Lie Algebras and the KillingForm
6 Decomposition of Symmetric Spaces
7 Rank and Classification
8 References
10 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Euclidean Space
Endow Rn with the Euclidean metric. The symmetry sp atp ∈ Rn is defined by
sp(p + v) := p − v .
Any line (i.e., geodesic) through p is of the form p + tv forsome v ∈ Rn, so sp reverse geodesics through p.Geometrically:
p
γ
sp
p
−γ
11 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Euclidean Space
Endow Rn with the Euclidean metric. The symmetry sp atp ∈ Rn is defined by
sp(p + v) := p − v .
Any line (i.e., geodesic) through p is of the form p + tv forsome v ∈ Rn, so sp reverse geodesics through p.Geometrically:
p
γ
sp
p
−γ
11 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Euclidean Space
Endow Rn with the Euclidean metric. The symmetry sp atp ∈ Rn is defined by
sp(p + v) := p − v .
Any line (i.e., geodesic) through p is of the form p + tv forsome v ∈ Rn, so sp reverse geodesics through p.Geometrically:
p
γ
sp
p
−γ
11 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Euclidean Space
Endow Rn with the Euclidean metric. The symmetry sp atp ∈ Rn is defined by
sp(p + v) := p − v .
Any line (i.e., geodesic) through p is of the form p + tv forsome v ∈ Rn, so sp reverse geodesics through p.Geometrically:
p
γ
sp
p
−γ
11 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Euclidean Space
Endow Rn with the Euclidean metric. The symmetry sp atp ∈ Rn is defined by
sp(p + v) := p − v .
Any line (i.e., geodesic) through p is of the form p + tv forsome v ∈ Rn, so sp reverse geodesics through p.Geometrically:
p
γ
sp
p
−γ
11 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Sphere
Endow the unit sphere Sn in Rn+1 with the metric inducedfrom the standard inner product. For p ∈ Sn, sp is thereflection about the line tp for t ∈ R in Rn+1. Precisely,
sp(q) := 2〈q, p〉p − q.
The symmetry sp reverse the direction of great circles throughp. Geometrically (for S2):
p
γsp
p
−γ
12 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Sphere
Endow the unit sphere Sn in Rn+1 with the metric inducedfrom the standard inner product. For p ∈ Sn, sp is thereflection about the line tp for t ∈ R in Rn+1. Precisely,
sp(q) := 2〈q, p〉p − q.
The symmetry sp reverse the direction of great circles throughp. Geometrically (for S2):
p
γsp
p
−γ
12 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Sphere
Endow the unit sphere Sn in Rn+1 with the metric inducedfrom the standard inner product. For p ∈ Sn, sp is thereflection about the line tp for t ∈ R in Rn+1. Precisely,
sp(q) := 2〈q, p〉p − q.
The symmetry sp reverse the direction of great circles throughp. Geometrically (for S2):
p
γsp
p
−γ
12 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Sphere
Endow the unit sphere Sn in Rn+1 with the metric inducedfrom the standard inner product. For p ∈ Sn, sp is thereflection about the line tp for t ∈ R in Rn+1. Precisely,
sp(q) := 2〈q, p〉p − q.
The symmetry sp reverse the direction of great circles throughp. Geometrically (for S2):
p
γsp
p
−γ
12 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Sphere
Endow the unit sphere Sn in Rn+1 with the metric inducedfrom the standard inner product. For p ∈ Sn, sp is thereflection about the line tp for t ∈ R in Rn+1. Precisely,
sp(q) := 2〈q, p〉p − q.
The symmetry sp reverse the direction of great circles throughp. Geometrically (for S2):
p
γsp
p
−γ
12 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Real Hyperbolic Space
To define Hn, give Rn+1 the Lorentzian scalar product definedby
〈p, q〉 :=
(n∑
i=1
piqi
)− pn+1qn+1,
and define Hn to be
Hn := {p ∈ Rn+1 | 〈p, p〉 = −1, pn+1 > 0}.
The induced scalar product on TpHn for p ∈ Hn makes Hn
into a Riemannian manifold. For any p ∈ Hn, the restriction of
sp(q) := 2〈q, p〉p − q
to Hn is the symmetry through p.
13 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Real Hyperbolic Space
To define Hn, give Rn+1 the Lorentzian scalar product definedby
〈p, q〉 :=
(n∑
i=1
piqi
)− pn+1qn+1,
and define Hn to be
Hn := {p ∈ Rn+1 | 〈p, p〉 = −1, pn+1 > 0}.
The induced scalar product on TpHn for p ∈ Hn makes Hn
into a Riemannian manifold. For any p ∈ Hn, the restriction of
sp(q) := 2〈q, p〉p − q
to Hn is the symmetry through p.
13 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Real Hyperbolic Space
To define Hn, give Rn+1 the Lorentzian scalar product definedby
〈p, q〉 :=
(n∑
i=1
piqi
)− pn+1qn+1,
and define Hn to be
Hn := {p ∈ Rn+1 | 〈p, p〉 = −1, pn+1 > 0}.
The induced scalar product on TpHn for p ∈ Hn makes Hn
into a Riemannian manifold. For any p ∈ Hn, the restriction of
sp(q) := 2〈q, p〉p − q
to Hn is the symmetry through p.
13 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Real Hyperbolic Space
To define Hn, give Rn+1 the Lorentzian scalar product definedby
〈p, q〉 :=
(n∑
i=1
piqi
)− pn+1qn+1,
and define Hn to be
Hn := {p ∈ Rn+1 | 〈p, p〉 = −1, pn+1 > 0}.
The induced scalar product on TpHn for p ∈ Hn makes Hn
into a Riemannian manifold. For any p ∈ Hn, the restriction of
sp(q) := 2〈q, p〉p − q
to Hn is the symmetry through p.
13 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Orthogonal Group
We first show O(n) is a homogeneous space. O(n) is a regularsubmanifold of GL(n,R) by the regular level set theorem. TheRiemannian structure on O(n) is induced from the scalarproduct on Rn. In particular,
〈A,B〉 := trace(ATB).
If G ∈ O(n), then
〈GA,GB〉 = (GA)TGB = ATGTGB = ATB = 〈A,B〉.
Similarly, 〈AG ,BG 〉 = 〈A,B〉. Therefore O(n) is ahomogeneous space since O(n) acts transitively on itself.
14 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Orthogonal Group
We first show O(n) is a homogeneous space. O(n) is a regularsubmanifold of GL(n,R) by the regular level set theorem. TheRiemannian structure on O(n) is induced from the scalarproduct on Rn. In particular,
〈A,B〉 := trace(ATB).
If G ∈ O(n), then
〈GA,GB〉 = (GA)TGB = ATGTGB = ATB = 〈A,B〉.
Similarly, 〈AG ,BG 〉 = 〈A,B〉. Therefore O(n) is ahomogeneous space since O(n) acts transitively on itself.
14 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Orthogonal Group
We first show O(n) is a homogeneous space. O(n) is a regularsubmanifold of GL(n,R) by the regular level set theorem. TheRiemannian structure on O(n) is induced from the scalarproduct on Rn. In particular,
〈A,B〉 := trace(ATB).
If G ∈ O(n), then
〈GA,GB〉 = (GA)TGB = ATGTGB = ATB = 〈A,B〉.
Similarly, 〈AG ,BG 〉 = 〈A,B〉. Therefore O(n) is ahomogeneous space since O(n) acts transitively on itself.
14 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Orthogonal Group
We first show O(n) is a homogeneous space. O(n) is a regularsubmanifold of GL(n,R) by the regular level set theorem. TheRiemannian structure on O(n) is induced from the scalarproduct on Rn. In particular,
〈A,B〉 := trace(ATB).
If G ∈ O(n), then
〈GA,GB〉 = (GA)TGB = ATGTGB = ATB = 〈A,B〉.
Similarly, 〈AG ,BG 〉 = 〈A,B〉. Therefore O(n) is ahomogeneous space since O(n) acts transitively on itself.
14 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Orthogonal Group
We first show O(n) is a homogeneous space. O(n) is a regularsubmanifold of GL(n,R) by the regular level set theorem. TheRiemannian structure on O(n) is induced from the scalarproduct on Rn. In particular,
〈A,B〉 := trace(ATB).
If G ∈ O(n), then
〈GA,GB〉 = (GA)TGB = ATGTGB = ATB = 〈A,B〉.
Similarly, 〈AG ,BG 〉 = 〈A,B〉. Therefore O(n) is ahomogeneous space since O(n) acts transitively on itself.
14 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Orthogonal Group
We first show O(n) is a homogeneous space. O(n) is a regularsubmanifold of GL(n,R) by the regular level set theorem. TheRiemannian structure on O(n) is induced from the scalarproduct on Rn. In particular,
〈A,B〉 := trace(ATB).
If G ∈ O(n), then
〈GA,GB〉 = (GA)TGB = ATGTGB = ATB = 〈A,B〉.
Similarly, 〈AG ,BG 〉 = 〈A,B〉. Therefore O(n) is ahomogeneous space since O(n) acts transitively on itself.
14 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Orthogonal Group
We first show O(n) is a homogeneous space. O(n) is a regularsubmanifold of GL(n,R) by the regular level set theorem. TheRiemannian structure on O(n) is induced from the scalarproduct on Rn. In particular,
〈A,B〉 := trace(ATB).
If G ∈ O(n), then
〈GA,GB〉 = (GA)TGB = ATGTGB = ATB = 〈A,B〉.
Similarly, 〈AG ,BG 〉 = 〈A,B〉. Therefore O(n) is ahomogeneous space since O(n) acts transitively on itself.
14 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Orthogonal Group
By Theorem 2.1 it suffices to exhibit a symmetry at the originI . Consider
sI : O(n)→ O(n) A 7→ AT .
sI is a isometry preserving the identity. It’s well-known (usingcurves)
TIO(n) = {A ∈ GL(n,R) | AT = −A}.
ComputingsI∗,I : TIO(n)→ TIO(n)
using curves, sI∗,I is the negative of the identity map. Thus sIis a symmetry at I .
15 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Orthogonal Group
By Theorem 2.1 it suffices to exhibit a symmetry at the originI . Consider
sI : O(n)→ O(n) A 7→ AT .
sI is a isometry preserving the identity. It’s well-known (usingcurves)
TIO(n) = {A ∈ GL(n,R) | AT = −A}.
ComputingsI∗,I : TIO(n)→ TIO(n)
using curves, sI∗,I is the negative of the identity map. Thus sIis a symmetry at I .
15 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Orthogonal Group
By Theorem 2.1 it suffices to exhibit a symmetry at the originI . Consider
sI : O(n)→ O(n) A 7→ AT .
sI is a isometry preserving the identity. It’s well-known (usingcurves)
TIO(n) = {A ∈ GL(n,R) | AT = −A}.
ComputingsI∗,I : TIO(n)→ TIO(n)
using curves, sI∗,I is the negative of the identity map. Thus sIis a symmetry at I .
15 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Orthogonal Group
By Theorem 2.1 it suffices to exhibit a symmetry at the originI . Consider
sI : O(n)→ O(n) A 7→ AT .
sI is a isometry preserving the identity. It’s well-known (usingcurves)
TIO(n) = {A ∈ GL(n,R) | AT = −A}.
ComputingsI∗,I : TIO(n)→ TIO(n)
using curves, sI∗,I is the negative of the identity map. Thus sIis a symmetry at I .
15 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Orthogonal Group
By Theorem 2.1 it suffices to exhibit a symmetry at the originI . Consider
sI : O(n)→ O(n) A 7→ AT .
sI is a isometry preserving the identity. It’s well-known (usingcurves)
TIO(n) = {A ∈ GL(n,R) | AT = −A}.
ComputingsI∗,I : TIO(n)→ TIO(n)
using curves, sI∗,I is the negative of the identity map. Thus sIis a symmetry at I .
15 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Orthogonal Group
By Theorem 2.1 it suffices to exhibit a symmetry at the originI . Consider
sI : O(n)→ O(n) A 7→ AT .
sI is a isometry preserving the identity. It’s well-known (usingcurves)
TIO(n) = {A ∈ GL(n,R) | AT = −A}.
ComputingsI∗,I : TIO(n)→ TIO(n)
using curves, sI∗,I is the negative of the identity map. Thus sIis a symmetry at I .
15 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Compact Lie Group
Any compact Lie group is a symmetric space. If G is acompact compact Lie group it exhibits a biinvariant metric. Gacts transitively on itself, implying G is homogeneous.Consider
se : G → G g 7→ g−1.
sI is a diffeomorphism preserving the identity, and se∗,epreserves the metric. If g ∈ G is arbitrary, notese ◦ `g = rg−1 ◦ se . By the chain rule
se∗,g ◦ `g∗,e = rg−1∗,e◦ se∗,e .
So se is an isometry, and hence a symmetry at e proving G isa symmetric space.This generalizes our discussion about O(n) .
16 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Compact Lie Group
Any compact Lie group is a symmetric space. If G is acompact compact Lie group it exhibits a biinvariant metric. Gacts transitively on itself, implying G is homogeneous.Consider
se : G → G g 7→ g−1.
sI is a diffeomorphism preserving the identity, and se∗,epreserves the metric. If g ∈ G is arbitrary, notese ◦ `g = rg−1 ◦ se . By the chain rule
se∗,g ◦ `g∗,e = rg−1∗,e◦ se∗,e .
So se is an isometry, and hence a symmetry at e proving G isa symmetric space.This generalizes our discussion about O(n) .
16 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Compact Lie Group
Any compact Lie group is a symmetric space. If G is acompact compact Lie group it exhibits a biinvariant metric. Gacts transitively on itself, implying G is homogeneous.Consider
se : G → G g 7→ g−1.
sI is a diffeomorphism preserving the identity, and se∗,epreserves the metric. If g ∈ G is arbitrary, notese ◦ `g = rg−1 ◦ se . By the chain rule
se∗,g ◦ `g∗,e = rg−1∗,e◦ se∗,e .
So se is an isometry, and hence a symmetry at e proving G isa symmetric space.This generalizes our discussion about O(n) .
16 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Compact Lie Group
Any compact Lie group is a symmetric space. If G is acompact compact Lie group it exhibits a biinvariant metric. Gacts transitively on itself, implying G is homogeneous.Consider
se : G → G g 7→ g−1.
sI is a diffeomorphism preserving the identity, and se∗,epreserves the metric. If g ∈ G is arbitrary, notese ◦ `g = rg−1 ◦ se . By the chain rule
se∗,g ◦ `g∗,e = rg−1∗,e◦ se∗,e .
So se is an isometry, and hence a symmetry at e proving G isa symmetric space.This generalizes our discussion about O(n) .
16 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Compact Lie Group
Any compact Lie group is a symmetric space. If G is acompact compact Lie group it exhibits a biinvariant metric. Gacts transitively on itself, implying G is homogeneous.Consider
se : G → G g 7→ g−1.
sI is a diffeomorphism preserving the identity, and se∗,epreserves the metric. If g ∈ G is arbitrary, notese ◦ `g = rg−1 ◦ se . By the chain rule
se∗,g ◦ `g∗,e = rg−1∗,e◦ se∗,e .
So se is an isometry, and hence a symmetry at e proving G isa symmetric space.This generalizes our discussion about O(n) .
16 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Compact Lie Group
Any compact Lie group is a symmetric space. If G is acompact compact Lie group it exhibits a biinvariant metric. Gacts transitively on itself, implying G is homogeneous.Consider
se : G → G g 7→ g−1.
sI is a diffeomorphism preserving the identity, and se∗,epreserves the metric. If g ∈ G is arbitrary, notese ◦ `g = rg−1 ◦ se . By the chain rule
se∗,g ◦ `g∗,e = rg−1∗,e◦ se∗,e .
So se is an isometry, and hence a symmetry at e proving G isa symmetric space.This generalizes our discussion about O(n) .
16 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Compact Lie Group
Any compact Lie group is a symmetric space. If G is acompact compact Lie group it exhibits a biinvariant metric. Gacts transitively on itself, implying G is homogeneous.Consider
se : G → G g 7→ g−1.
sI is a diffeomorphism preserving the identity, and se∗,epreserves the metric. If g ∈ G is arbitrary, notese ◦ `g = rg−1 ◦ se . By the chain rule
se∗,g ◦ `g∗,e = rg−1∗,e◦ se∗,e .
So se is an isometry, and hence a symmetry at e proving G isa symmetric space.This generalizes our discussion about O(n) .
16 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Compact Lie Group
Any compact Lie group is a symmetric space. If G is acompact compact Lie group it exhibits a biinvariant metric. Gacts transitively on itself, implying G is homogeneous.Consider
se : G → G g 7→ g−1.
sI is a diffeomorphism preserving the identity, and se∗,epreserves the metric. If g ∈ G is arbitrary, notese ◦ `g = rg−1 ◦ se . By the chain rule
se∗,g ◦ `g∗,e = rg−1∗,e◦ se∗,e .
So se is an isometry, and hence a symmetry at e proving G isa symmetric space.This generalizes our discussion about O(n) .
16 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Compact Lie Group
Any compact Lie group is a symmetric space. If G is acompact compact Lie group it exhibits a biinvariant metric. Gacts transitively on itself, implying G is homogeneous.Consider
se : G → G g 7→ g−1.
sI is a diffeomorphism preserving the identity, and se∗,epreserves the metric. If g ∈ G is arbitrary, notese ◦ `g = rg−1 ◦ se . By the chain rule
se∗,g ◦ `g∗,e = rg−1∗,e◦ se∗,e .
So se is an isometry, and hence a symmetry at e proving G isa symmetric space.This generalizes our discussion about O(n) .
16 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Outline
1 History
2 Symmetric Spaces
3 Examples of Symmetric Spaces
4 Curvature and Locally Symmetric Spaces
5 Effective Orthogonal Symmetric Lie Algebras and the KillingForm
6 Decomposition of Symmetric Spaces
7 Rank and Classification
8 References
17 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Parallel Curvature Tensor
Symmetric space have parallel curvature tensor.
Theorem 4.1
If M is a symmetric space with curvature tensor R, then thecurvature tensor is parallel, i.e., ∇R = 0.
Proof sketch. We prove ∇R is locally parallel. Note that if Tis a covariant k-tensor in a vector space which is invariantunder −id, then T = (−1)kT . If k is odd then necessarilyT = 0. If M is symmetric, each point p ∈ M admits asymmetry sp. We then check (∇R)p is invariant under sp∗,pand has odd rank.
18 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Parallel Curvature Tensor
Symmetric space have parallel curvature tensor.
Theorem 4.1
If M is a symmetric space with curvature tensor R, then thecurvature tensor is parallel, i.e., ∇R = 0.
Proof sketch. We prove ∇R is locally parallel. Note that if Tis a covariant k-tensor in a vector space which is invariantunder −id, then T = (−1)kT . If k is odd then necessarilyT = 0. If M is symmetric, each point p ∈ M admits asymmetry sp. We then check (∇R)p is invariant under sp∗,pand has odd rank.
18 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Parallel Curvature Tensor
Symmetric space have parallel curvature tensor.
Theorem 4.1
If M is a symmetric space with curvature tensor R, then thecurvature tensor is parallel, i.e., ∇R = 0.
Proof sketch. We prove ∇R is locally parallel. Note that if Tis a covariant k-tensor in a vector space which is invariantunder −id, then T = (−1)kT . If k is odd then necessarilyT = 0. If M is symmetric, each point p ∈ M admits asymmetry sp. We then check (∇R)p is invariant under sp∗,pand has odd rank.
18 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Parallel Curvature Tensor
Symmetric space have parallel curvature tensor.
Theorem 4.1
If M is a symmetric space with curvature tensor R, then thecurvature tensor is parallel, i.e., ∇R = 0.
Proof sketch. We prove ∇R is locally parallel. Note that if Tis a covariant k-tensor in a vector space which is invariantunder −id, then T = (−1)kT . If k is odd then necessarilyT = 0. If M is symmetric, each point p ∈ M admits asymmetry sp. We then check (∇R)p is invariant under sp∗,pand has odd rank.
18 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Parallel Curvature Tensor
Symmetric space have parallel curvature tensor.
Theorem 4.1
If M is a symmetric space with curvature tensor R, then thecurvature tensor is parallel, i.e., ∇R = 0.
Proof sketch. We prove ∇R is locally parallel. Note that if Tis a covariant k-tensor in a vector space which is invariantunder −id, then T = (−1)kT . If k is odd then necessarilyT = 0. If M is symmetric, each point p ∈ M admits asymmetry sp. We then check (∇R)p is invariant under sp∗,pand has odd rank.
18 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Parallel Curvature Tensor
Symmetric space have parallel curvature tensor.
Theorem 4.1
If M is a symmetric space with curvature tensor R, then thecurvature tensor is parallel, i.e., ∇R = 0.
Proof sketch. We prove ∇R is locally parallel. Note that if Tis a covariant k-tensor in a vector space which is invariantunder −id, then T = (−1)kT . If k is odd then necessarilyT = 0. If M is symmetric, each point p ∈ M admits asymmetry sp. We then check (∇R)p is invariant under sp∗,pand has odd rank.
18 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Parallel Curvature Tensor
Symmetric space have parallel curvature tensor.
Theorem 4.1
If M is a symmetric space with curvature tensor R, then thecurvature tensor is parallel, i.e., ∇R = 0.
Proof sketch. We prove ∇R is locally parallel. Note that if Tis a covariant k-tensor in a vector space which is invariantunder −id, then T = (−1)kT . If k is odd then necessarilyT = 0. If M is symmetric, each point p ∈ M admits asymmetry sp. We then check (∇R)p is invariant under sp∗,pand has odd rank.
18 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Locally Symmetric Spaces
Definition 4.1
A Riemannian manifold M is a locally symmetric space if ithas parallel curvature tensor.
A theorem of Cartan justifies this definition:
Theorem 4.2
If M is a locally symmetric space then for each p ∈ M, there isa symmetry at p defined in a neighborhood of p. Moreover, ifM is simply connected and complete, M is a symmetric space.
19 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Locally Symmetric Spaces
Definition 4.1
A Riemannian manifold M is a locally symmetric space if ithas parallel curvature tensor.
A theorem of Cartan justifies this definition:
Theorem 4.2
If M is a locally symmetric space then for each p ∈ M, there isa symmetry at p defined in a neighborhood of p. Moreover, ifM is simply connected and complete, M is a symmetric space.
19 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Locally Symmetric Spaces
Definition 4.1
A Riemannian manifold M is a locally symmetric space if ithas parallel curvature tensor.
A theorem of Cartan justifies this definition:
Theorem 4.2
If M is a locally symmetric space then for each p ∈ M, there isa symmetry at p defined in a neighborhood of p. Moreover, ifM is simply connected and complete, M is a symmetric space.
19 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Locally Symmetric Spaces
Definition 4.1
A Riemannian manifold M is a locally symmetric space if ithas parallel curvature tensor.
A theorem of Cartan justifies this definition:
Theorem 4.2
If M is a locally symmetric space then for each p ∈ M, there isa symmetry at p defined in a neighborhood of p. Moreover, ifM is simply connected and complete, M is a symmetric space.
19 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Locally Symmetric Spaces
Proof sketch. For ε > 0, the exponential
expp : B(0, ε)→ B(p, ε)
is a diffeomorphism. Under exp−1p we define sp(x) := −x .
This induces a diffeomorphism
sp : B(p, ε)→ B(p, ε),
and sp∗,p = −id on TpM automatically. We use the parallelcurvature tensor to prove sp is an isometry. The secondstatement is proved using an analytic continuation.
20 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Locally Symmetric Spaces
Proof sketch. For ε > 0, the exponential
expp : B(0, ε)→ B(p, ε)
is a diffeomorphism. Under exp−1p we define sp(x) := −x .
This induces a diffeomorphism
sp : B(p, ε)→ B(p, ε),
and sp∗,p = −id on TpM automatically. We use the parallelcurvature tensor to prove sp is an isometry. The secondstatement is proved using an analytic continuation.
20 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Locally Symmetric Spaces
Proof sketch. For ε > 0, the exponential
expp : B(0, ε)→ B(p, ε)
is a diffeomorphism. Under exp−1p we define sp(x) := −x .
This induces a diffeomorphism
sp : B(p, ε)→ B(p, ε),
and sp∗,p = −id on TpM automatically. We use the parallelcurvature tensor to prove sp is an isometry. The secondstatement is proved using an analytic continuation.
20 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Locally Symmetric Spaces
Proof sketch. For ε > 0, the exponential
expp : B(0, ε)→ B(p, ε)
is a diffeomorphism. Under exp−1p we define sp(x) := −x .
This induces a diffeomorphism
sp : B(p, ε)→ B(p, ε),
and sp∗,p = −id on TpM automatically. We use the parallelcurvature tensor to prove sp is an isometry. The secondstatement is proved using an analytic continuation.
20 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Locally Symmetric Spaces
Proof sketch. For ε > 0, the exponential
expp : B(0, ε)→ B(p, ε)
is a diffeomorphism. Under exp−1p we define sp(x) := −x .
This induces a diffeomorphism
sp : B(p, ε)→ B(p, ε),
and sp∗,p = −id on TpM automatically. We use the parallelcurvature tensor to prove sp is an isometry. The secondstatement is proved using an analytic continuation.
20 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Outline
1 History
2 Symmetric Spaces
3 Examples of Symmetric Spaces
4 Curvature and Locally Symmetric Spaces
5 Effective Orthogonal Symmetric Lie Algebras and the KillingForm
6 Decomposition of Symmetric Spaces
7 Rank and Classification
8 References
21 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Orthogonal Symmetric Lie Algebras
Definition 5.1
An orthogonal symmetric Lie algebra (g, s) is a Lie algebra gand an involution automorphism s of g such that theeigenspace u of s corresponding to 1 (i.e., the set of fixedpoints of s) is a compact Lie subalgebra. An orthogonalsymmetric Lie algebra is effective if u and Z (g) intersecttrivially.
As a prototypical example, let g = R and s = −id so thatu = {0}. Then (g, s) is an effective orthogonal symmetric Liealgebra.
22 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Orthogonal Symmetric Lie Algebras
Definition 5.1
An orthogonal symmetric Lie algebra (g, s) is a Lie algebra gand an involution automorphism s of g such that theeigenspace u of s corresponding to 1 (i.e., the set of fixedpoints of s) is a compact Lie subalgebra. An orthogonalsymmetric Lie algebra is effective if u and Z (g) intersecttrivially.
As a prototypical example, let g = R and s = −id so thatu = {0}. Then (g, s) is an effective orthogonal symmetric Liealgebra.
22 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Orthogonal Symmetric Lie Algebras
Definition 5.1
An orthogonal symmetric Lie algebra (g, s) is a Lie algebra gand an involution automorphism s of g such that theeigenspace u of s corresponding to 1 (i.e., the set of fixedpoints of s) is a compact Lie subalgebra. An orthogonalsymmetric Lie algebra is effective if u and Z (g) intersecttrivially.
As a prototypical example, let g = R and s = −id so thatu = {0}. Then (g, s) is an effective orthogonal symmetric Liealgebra.
22 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Orthogonal Symmetric Lie Algebras
Definition 5.1
An orthogonal symmetric Lie algebra (g, s) is a Lie algebra gand an involution automorphism s of g such that theeigenspace u of s corresponding to 1 (i.e., the set of fixedpoints of s) is a compact Lie subalgebra. An orthogonalsymmetric Lie algebra is effective if u and Z (g) intersecttrivially.
As a prototypical example, let g = R and s = −id so thatu = {0}. Then (g, s) is an effective orthogonal symmetric Liealgebra.
22 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Killing Form
Definition 5.2
If g is a Lie algebra, we define the Killing form B of g over afield F to be the bilinear form
B : g⊗ g→ F x ⊗ y 7→ trace(Ad(x) ◦Ad(y)).
Usually F = R. The Killing form is symmetric and satisfiesother nice property. The sign of the Killing form plays animportant role in Cartan’s classification.
23 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Killing Form
Definition 5.2
If g is a Lie algebra, we define the Killing form B of g over afield F to be the bilinear form
B : g⊗ g→ F x ⊗ y 7→ trace(Ad(x) ◦Ad(y)).
Usually F = R. The Killing form is symmetric and satisfiesother nice property. The sign of the Killing form plays animportant role in Cartan’s classification.
23 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Killing Form
Definition 5.2
If g is a Lie algebra, we define the Killing form B of g over afield F to be the bilinear form
B : g⊗ g→ F x ⊗ y 7→ trace(Ad(x) ◦Ad(y)).
Usually F = R. The Killing form is symmetric and satisfiesother nice property. The sign of the Killing form plays animportant role in Cartan’s classification.
23 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Killing Form
Definition 5.2
If g is a Lie algebra, we define the Killing form B of g over afield F to be the bilinear form
B : g⊗ g→ F x ⊗ y 7→ trace(Ad(x) ◦Ad(y)).
Usually F = R. The Killing form is symmetric and satisfiesother nice property. The sign of the Killing form plays animportant role in Cartan’s classification.
23 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Types
Definition 5.3
Let g be and orthogonal symmetric Lie algebra.
• We say g is of compact type if B is negative definite.
• We say g is of noncompact type if B is positive definite.
• We say g is of flat type if B is identically zero.
Lie algebras of compact type are compact, and Lie algebras ofnoncompact type are noncompact.
24 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Types
Definition 5.3
Let g be and orthogonal symmetric Lie algebra.
• We say g is of compact type if B is negative definite.
• We say g is of noncompact type if B is positive definite.
• We say g is of flat type if B is identically zero.
Lie algebras of compact type are compact, and Lie algebras ofnoncompact type are noncompact.
24 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Types
Definition 5.3
Let g be and orthogonal symmetric Lie algebra.
• We say g is of compact type if B is negative definite.
• We say g is of noncompact type if B is positive definite.
• We say g is of flat type if B is identically zero.
Lie algebras of compact type are compact, and Lie algebras ofnoncompact type are noncompact.
24 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Types
Definition 5.3
Let g be and orthogonal symmetric Lie algebra.
• We say g is of compact type if B is negative definite.
• We say g is of noncompact type if B is positive definite.
• We say g is of flat type if B is identically zero.
Lie algebras of compact type are compact, and Lie algebras ofnoncompact type are noncompact.
24 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Types
Definition 5.3
Let g be and orthogonal symmetric Lie algebra.
• We say g is of compact type if B is negative definite.
• We say g is of noncompact type if B is positive definite.
• We say g is of flat type if B is identically zero.
Lie algebras of compact type are compact, and Lie algebras ofnoncompact type are noncompact.
24 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Natural Decomposition
Definition 5.3 is useful when g is effective.
Theorem 5.1
Let g be an effective orthogonal symmetric Lie algebra. Theng admits the mutually orthogonal decomposition
g = g0 ⊕ g+ ⊕ g−
where g0 is of flat type, g+ is of compact type, and g− is ofnoncompact type.
The astonishing fact is that (simply connected) symmetricspaces decompose this way.
25 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Natural Decomposition
Definition 5.3 is useful when g is effective.
Theorem 5.1
Let g be an effective orthogonal symmetric Lie algebra. Theng admits the mutually orthogonal decomposition
g = g0 ⊕ g+ ⊕ g−
where g0 is of flat type, g+ is of compact type, and g− is ofnoncompact type.
The astonishing fact is that (simply connected) symmetricspaces decompose this way.
25 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Natural Decomposition
Definition 5.3 is useful when g is effective.
Theorem 5.1
Let g be an effective orthogonal symmetric Lie algebra. Theng admits the mutually orthogonal decomposition
g = g0 ⊕ g+ ⊕ g−
where g0 is of flat type, g+ is of compact type, and g− is ofnoncompact type.
The astonishing fact is that (simply connected) symmetricspaces decompose this way.
25 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
The Natural Decomposition
Definition 5.3 is useful when g is effective.
Theorem 5.1
Let g be an effective orthogonal symmetric Lie algebra. Theng admits the mutually orthogonal decomposition
g = g0 ⊕ g+ ⊕ g−
where g0 is of flat type, g+ is of compact type, and g− is ofnoncompact type.
The astonishing fact is that (simply connected) symmetricspaces decompose this way.
25 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Riemannian Symmetric Pairs
Definition 5.4
Let G be a connected Lie group with K ≤ G a closedsubgroup. We say (G ,K ) is a Riemannian symmetric pair ifthe following two properties are satisfied:
1 AdG (K ) ≤ GL(g) is compact.
2 There exists an involution σ : G → G such that(Gσ)◦ ⊆ K ⊆ Gσ.
If M is a symmetric space, (Iso(M)◦, Iso(M)p) is aRiemannian symmetric pair with
σ : Iso(M)◦ → Iso(M)◦ s 7→ sp ◦ s ◦ s−1p .
Given a Riemannian symmetric pair (G ,K ), G/K is asymmetric space with respect to any G -invariant Riemannianmetric.
26 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Riemannian Symmetric Pairs
Definition 5.4
Let G be a connected Lie group with K ≤ G a closedsubgroup. We say (G ,K ) is a Riemannian symmetric pair ifthe following two properties are satisfied:
1 AdG (K ) ≤ GL(g) is compact.
2 There exists an involution σ : G → G such that(Gσ)◦ ⊆ K ⊆ Gσ.
If M is a symmetric space, (Iso(M)◦, Iso(M)p) is aRiemannian symmetric pair with
σ : Iso(M)◦ → Iso(M)◦ s 7→ sp ◦ s ◦ s−1p .
Given a Riemannian symmetric pair (G ,K ), G/K is asymmetric space with respect to any G -invariant Riemannianmetric.
26 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Riemannian Symmetric Pairs
Definition 5.4
Let G be a connected Lie group with K ≤ G a closedsubgroup. We say (G ,K ) is a Riemannian symmetric pair ifthe following two properties are satisfied:
1 AdG (K ) ≤ GL(g) is compact.
2 There exists an involution σ : G → G such that(Gσ)◦ ⊆ K ⊆ Gσ.
If M is a symmetric space, (Iso(M)◦, Iso(M)p) is aRiemannian symmetric pair with
σ : Iso(M)◦ → Iso(M)◦ s 7→ sp ◦ s ◦ s−1p .
Given a Riemannian symmetric pair (G ,K ), G/K is asymmetric space with respect to any G -invariant Riemannianmetric.
26 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Riemannian Symmetric Pairs
Definition 5.4
Let G be a connected Lie group with K ≤ G a closedsubgroup. We say (G ,K ) is a Riemannian symmetric pair ifthe following two properties are satisfied:
1 AdG (K ) ≤ GL(g) is compact.
2 There exists an involution σ : G → G such that(Gσ)◦ ⊆ K ⊆ Gσ.
If M is a symmetric space, (Iso(M)◦, Iso(M)p) is aRiemannian symmetric pair with
σ : Iso(M)◦ → Iso(M)◦ s 7→ sp ◦ s ◦ s−1p .
Given a Riemannian symmetric pair (G ,K ), G/K is asymmetric space with respect to any G -invariant Riemannianmetric.
26 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Riemannian Symmetric Pairs
Definition 5.4
Let G be a connected Lie group with K ≤ G a closedsubgroup. We say (G ,K ) is a Riemannian symmetric pair ifthe following two properties are satisfied:
1 AdG (K ) ≤ GL(g) is compact.
2 There exists an involution σ : G → G such that(Gσ)◦ ⊆ K ⊆ Gσ.
If M is a symmetric space, (Iso(M)◦, Iso(M)p) is aRiemannian symmetric pair with
σ : Iso(M)◦ → Iso(M)◦ s 7→ sp ◦ s ◦ s−1p .
Given a Riemannian symmetric pair (G ,K ), G/K is asymmetric space with respect to any G -invariant Riemannianmetric.
26 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Riemannian Symmetric Pairs
Definition 5.4
Let G be a connected Lie group with K ≤ G a closedsubgroup. We say (G ,K ) is a Riemannian symmetric pair ifthe following two properties are satisfied:
1 AdG (K ) ≤ GL(g) is compact.
2 There exists an involution σ : G → G such that(Gσ)◦ ⊆ K ⊆ Gσ.
If M is a symmetric space, (Iso(M)◦, Iso(M)p) is aRiemannian symmetric pair with
σ : Iso(M)◦ → Iso(M)◦ s 7→ sp ◦ s ◦ s−1p .
Given a Riemannian symmetric pair (G ,K ), G/K is asymmetric space with respect to any G -invariant Riemannianmetric.
26 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Correspondences
Given a Riemannian symmetric pair (G ,K ), let g be the Liealgebra of G and set s = σ∗,e . Then (g, s) is a orthogonalsymmetric Lie algebra.
Definition 5.5
A Riemannian symmetric pair (G ,K ) is effective if Z (G ) ∩ Kis a discrete subgroup of G .
This is equivalent to (g, s) being effective. We know everysymmetric space gives rise to a Riemannian symmetric pair;this pair is always effective.
27 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Correspondences
Given a Riemannian symmetric pair (G ,K ), let g be the Liealgebra of G and set s = σ∗,e . Then (g, s) is a orthogonalsymmetric Lie algebra.
Definition 5.5
A Riemannian symmetric pair (G ,K ) is effective if Z (G ) ∩ Kis a discrete subgroup of G .
This is equivalent to (g, s) being effective. We know everysymmetric space gives rise to a Riemannian symmetric pair;this pair is always effective.
27 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Correspondences
Given a Riemannian symmetric pair (G ,K ), let g be the Liealgebra of G and set s = σ∗,e . Then (g, s) is a orthogonalsymmetric Lie algebra.
Definition 5.5
A Riemannian symmetric pair (G ,K ) is effective if Z (G ) ∩ Kis a discrete subgroup of G .
This is equivalent to (g, s) being effective. We know everysymmetric space gives rise to a Riemannian symmetric pair;this pair is always effective.
27 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Correspondences
Given a Riemannian symmetric pair (G ,K ), let g be the Liealgebra of G and set s = σ∗,e . Then (g, s) is a orthogonalsymmetric Lie algebra.
Definition 5.5
A Riemannian symmetric pair (G ,K ) is effective if Z (G ) ∩ Kis a discrete subgroup of G .
This is equivalent to (g, s) being effective. We know everysymmetric space gives rise to a Riemannian symmetric pair;this pair is always effective.
27 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Correspondences
Given a Riemannian symmetric pair (G ,K ), let g be the Liealgebra of G and set s = σ∗,e . Then (g, s) is a orthogonalsymmetric Lie algebra.
Definition 5.5
A Riemannian symmetric pair (G ,K ) is effective if Z (G ) ∩ Kis a discrete subgroup of G .
This is equivalent to (g, s) being effective. We know everysymmetric space gives rise to a Riemannian symmetric pair;this pair is always effective.
27 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Correspondences
Given a Riemannian symmetric pair (G ,K ), let g be the Liealgebra of G and set s = σ∗,e . Then (g, s) is a orthogonalsymmetric Lie algebra.
Definition 5.5
A Riemannian symmetric pair (G ,K ) is effective if Z (G ) ∩ Kis a discrete subgroup of G .
This is equivalent to (g, s) being effective. We know everysymmetric space gives rise to a Riemannian symmetric pair;this pair is always effective.
27 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Types Revisited
Definition 5.6
1 An effective Riemannian symmetric pair (G ,K ) is of flat,compact, or noncompact type if the correspondingeffective orthogonal symmetric Lie algebra (g, s) is offlat, compact, or noncompact type.
2 A symmetric space M is of flat, compact, or noncompacttype if the corresponding effective orthogonal symmetricLie algebra is of flat, compact, or noncompact type.
It is the second of these two conventions that we will makeuse of.
28 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Types Revisited
Definition 5.6
1 An effective Riemannian symmetric pair (G ,K ) is of flat,compact, or noncompact type if the correspondingeffective orthogonal symmetric Lie algebra (g, s) is offlat, compact, or noncompact type.
2 A symmetric space M is of flat, compact, or noncompacttype if the corresponding effective orthogonal symmetricLie algebra is of flat, compact, or noncompact type.
It is the second of these two conventions that we will makeuse of.
28 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Types Revisited
Definition 5.6
1 An effective Riemannian symmetric pair (G ,K ) is of flat,compact, or noncompact type if the correspondingeffective orthogonal symmetric Lie algebra (g, s) is offlat, compact, or noncompact type.
2 A symmetric space M is of flat, compact, or noncompacttype if the corresponding effective orthogonal symmetricLie algebra is of flat, compact, or noncompact type.
It is the second of these two conventions that we will makeuse of.
28 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Outline
1 History
2 Symmetric Spaces
3 Examples of Symmetric Spaces
4 Curvature and Locally Symmetric Spaces
5 Effective Orthogonal Symmetric Lie Algebras and the KillingForm
6 Decomposition of Symmetric Spaces
7 Rank and Classification
8 References
29 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Symmetric Space Decomposition
Observe that products of symmetric spaces are symmetric.With this observation, Cartan was able to prove the followingtheorem:
Theorem 6.1
A simply connected symmetric space M admits adecomposition
M ∼= M0 ×M+ ×M−
into symmetric spaces where M0 is of flat type, M+ is ofcompact type, and M− is of noncompact type.
30 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Symmetric Space Decomposition
Observe that products of symmetric spaces are symmetric.With this observation, Cartan was able to prove the followingtheorem:
Theorem 6.1
A simply connected symmetric space M admits adecomposition
M ∼= M0 ×M+ ×M−
into symmetric spaces where M0 is of flat type, M+ is ofcompact type, and M− is of noncompact type.
30 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Symmetric Space Decomposition
Observe that products of symmetric spaces are symmetric.With this observation, Cartan was able to prove the followingtheorem:
Theorem 6.1
A simply connected symmetric space M admits adecomposition
M ∼= M0 ×M+ ×M−
into symmetric spaces where M0 is of flat type, M+ is ofcompact type, and M− is of noncompact type.
30 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Decomposition of Symmetric Spaces
Proof sketch. By Theorem 2.2 M ∼= Iso(M)◦/Iso(M)p byfixing a basepoint p ∈ M. Recall (Iso(M)◦, Iso(M)p) is aRiemannian symmetric pair. Let (g, s) be the correspondingeffective orthogonal symmetric Lie algebra. By Theorem 5.1
g = g0 ⊕ g+ ⊕ g−.
Let G̃0, G̃+, and G̃− be the Lie groups which are the coveringspaces of the Lie groups associated to the Lie algebras above.Let K0, K+, and K− be the Lie groups corresponding to theLie subalgebras u0, u+, and u−. Then check
M ∼= (G̃0/K0)× (G̃+/K+)× (G̃−/K−),
where G̃0/K0 is of flat type, G̃+/K+ is of compact type, andG̃−/K− is of noncompact type.
31 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Decomposition of Symmetric Spaces
Proof sketch. By Theorem 2.2 M ∼= Iso(M)◦/Iso(M)p byfixing a basepoint p ∈ M. Recall (Iso(M)◦, Iso(M)p) is aRiemannian symmetric pair. Let (g, s) be the correspondingeffective orthogonal symmetric Lie algebra. By Theorem 5.1
g = g0 ⊕ g+ ⊕ g−.
Let G̃0, G̃+, and G̃− be the Lie groups which are the coveringspaces of the Lie groups associated to the Lie algebras above.Let K0, K+, and K− be the Lie groups corresponding to theLie subalgebras u0, u+, and u−. Then check
M ∼= (G̃0/K0)× (G̃+/K+)× (G̃−/K−),
where G̃0/K0 is of flat type, G̃+/K+ is of compact type, andG̃−/K− is of noncompact type.
31 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Decomposition of Symmetric Spaces
Proof sketch. By Theorem 2.2 M ∼= Iso(M)◦/Iso(M)p byfixing a basepoint p ∈ M. Recall (Iso(M)◦, Iso(M)p) is aRiemannian symmetric pair. Let (g, s) be the correspondingeffective orthogonal symmetric Lie algebra. By Theorem 5.1
g = g0 ⊕ g+ ⊕ g−.
Let G̃0, G̃+, and G̃− be the Lie groups which are the coveringspaces of the Lie groups associated to the Lie algebras above.Let K0, K+, and K− be the Lie groups corresponding to theLie subalgebras u0, u+, and u−. Then check
M ∼= (G̃0/K0)× (G̃+/K+)× (G̃−/K−),
where G̃0/K0 is of flat type, G̃+/K+ is of compact type, andG̃−/K− is of noncompact type.
31 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Decomposition of Symmetric Spaces
Proof sketch. By Theorem 2.2 M ∼= Iso(M)◦/Iso(M)p byfixing a basepoint p ∈ M. Recall (Iso(M)◦, Iso(M)p) is aRiemannian symmetric pair. Let (g, s) be the correspondingeffective orthogonal symmetric Lie algebra. By Theorem 5.1
g = g0 ⊕ g+ ⊕ g−.
Let G̃0, G̃+, and G̃− be the Lie groups which are the coveringspaces of the Lie groups associated to the Lie algebras above.Let K0, K+, and K− be the Lie groups corresponding to theLie subalgebras u0, u+, and u−. Then check
M ∼= (G̃0/K0)× (G̃+/K+)× (G̃−/K−),
where G̃0/K0 is of flat type, G̃+/K+ is of compact type, andG̃−/K− is of noncompact type.
31 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Decomposition of Symmetric Spaces
Proof sketch. By Theorem 2.2 M ∼= Iso(M)◦/Iso(M)p byfixing a basepoint p ∈ M. Recall (Iso(M)◦, Iso(M)p) is aRiemannian symmetric pair. Let (g, s) be the correspondingeffective orthogonal symmetric Lie algebra. By Theorem 5.1
g = g0 ⊕ g+ ⊕ g−.
Let G̃0, G̃+, and G̃− be the Lie groups which are the coveringspaces of the Lie groups associated to the Lie algebras above.Let K0, K+, and K− be the Lie groups corresponding to theLie subalgebras u0, u+, and u−. Then check
M ∼= (G̃0/K0)× (G̃+/K+)× (G̃−/K−),
where G̃0/K0 is of flat type, G̃+/K+ is of compact type, andG̃−/K− is of noncompact type.
31 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Decomposition of Symmetric Spaces
Proof sketch. By Theorem 2.2 M ∼= Iso(M)◦/Iso(M)p byfixing a basepoint p ∈ M. Recall (Iso(M)◦, Iso(M)p) is aRiemannian symmetric pair. Let (g, s) be the correspondingeffective orthogonal symmetric Lie algebra. By Theorem 5.1
g = g0 ⊕ g+ ⊕ g−.
Let G̃0, G̃+, and G̃− be the Lie groups which are the coveringspaces of the Lie groups associated to the Lie algebras above.Let K0, K+, and K− be the Lie groups corresponding to theLie subalgebras u0, u+, and u−. Then check
M ∼= (G̃0/K0)× (G̃+/K+)× (G̃−/K−),
where G̃0/K0 is of flat type, G̃+/K+ is of compact type, andG̃−/K− is of noncompact type.
31 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Comments
The assumption M is simply connected can be made withoutloss of generality. If we assume M is irreducible, i.e., not aproduct of symmetric spaces, then Theorem 6.1 says M iseither of compact, noncompact, or Euclidean type. So, itsuffices to classify symmetric spaces of these types. In orderto do so we will need an invariant: the rank of a symmetricspace.
32 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Comments
The assumption M is simply connected can be made withoutloss of generality. If we assume M is irreducible, i.e., not aproduct of symmetric spaces, then Theorem 6.1 says M iseither of compact, noncompact, or Euclidean type. So, itsuffices to classify symmetric spaces of these types. In orderto do so we will need an invariant: the rank of a symmetricspace.
32 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Comments
The assumption M is simply connected can be made withoutloss of generality. If we assume M is irreducible, i.e., not aproduct of symmetric spaces, then Theorem 6.1 says M iseither of compact, noncompact, or Euclidean type. So, itsuffices to classify symmetric spaces of these types. In orderto do so we will need an invariant: the rank of a symmetricspace.
32 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Comments
The assumption M is simply connected can be made withoutloss of generality. If we assume M is irreducible, i.e., not aproduct of symmetric spaces, then Theorem 6.1 says M iseither of compact, noncompact, or Euclidean type. So, itsuffices to classify symmetric spaces of these types. In orderto do so we will need an invariant: the rank of a symmetricspace.
32 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Comments
The assumption M is simply connected can be made withoutloss of generality. If we assume M is irreducible, i.e., not aproduct of symmetric spaces, then Theorem 6.1 says M iseither of compact, noncompact, or Euclidean type. So, itsuffices to classify symmetric spaces of these types. In orderto do so we will need an invariant: the rank of a symmetricspace.
32 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Outline
1 History
2 Symmetric Spaces
3 Examples of Symmetric Spaces
4 Curvature and Locally Symmetric Spaces
5 Effective Orthogonal Symmetric Lie Algebras and the KillingForm
6 Decomposition of Symmetric Spaces
7 Rank and Classification
8 References
33 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Flats and Rank
Definition 7.1
Suppose M is an irreducible symmetric space. A totallygeodesic immersion of Rn into M is called a flat. A flat ismaximal if it is not contained in any larger flat.
Basic Lie theory shows that all maximal flats of M are of thesame dimension.
Definition 7.2
The rank of and irreducible symmetric space M is thedimension of any maximal flat.
The rank of a symmetric space plays a very important role inCartan’s classification
34 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Flats and Rank
Definition 7.1
Suppose M is an irreducible symmetric space. A totallygeodesic immersion of Rn into M is called a flat. A flat ismaximal if it is not contained in any larger flat.
Basic Lie theory shows that all maximal flats of M are of thesame dimension.
Definition 7.2
The rank of and irreducible symmetric space M is thedimension of any maximal flat.
The rank of a symmetric space plays a very important role inCartan’s classification
34 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Flats and Rank
Definition 7.1
Suppose M is an irreducible symmetric space. A totallygeodesic immersion of Rn into M is called a flat. A flat ismaximal if it is not contained in any larger flat.
Basic Lie theory shows that all maximal flats of M are of thesame dimension.
Definition 7.2
The rank of and irreducible symmetric space M is thedimension of any maximal flat.
The rank of a symmetric space plays a very important role inCartan’s classification
34 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Flats and Rank
Definition 7.1
Suppose M is an irreducible symmetric space. A totallygeodesic immersion of Rn into M is called a flat. A flat ismaximal if it is not contained in any larger flat.
Basic Lie theory shows that all maximal flats of M are of thesame dimension.
Definition 7.2
The rank of and irreducible symmetric space M is thedimension of any maximal flat.
The rank of a symmetric space plays a very important role inCartan’s classification
34 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Flats and Rank
Definition 7.1
Suppose M is an irreducible symmetric space. A totallygeodesic immersion of Rn into M is called a flat. A flat ismaximal if it is not contained in any larger flat.
Basic Lie theory shows that all maximal flats of M are of thesame dimension.
Definition 7.2
The rank of and irreducible symmetric space M is thedimension of any maximal flat.
The rank of a symmetric space plays a very important role inCartan’s classification
34 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Flats and Rank
Definition 7.1
Suppose M is an irreducible symmetric space. A totallygeodesic immersion of Rn into M is called a flat. A flat ismaximal if it is not contained in any larger flat.
Basic Lie theory shows that all maximal flats of M are of thesame dimension.
Definition 7.2
The rank of and irreducible symmetric space M is thedimension of any maximal flat.
The rank of a symmetric space plays a very important role inCartan’s classification
34 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Rank and Sectional Curvature
The rank of is at least one with equality if the sectionalcurvature is positive or negative. If the sectional curvature ispositive the space is of compact type, and if the sectionalcurvature is negative the space is of noncompact type.The rank of a Euclidean type space is equal to its dimension.This implies Euclidean type spaces are isometric to Euclideanspace of that dimension.Therefore we are reduces to classifying symmetric spaces ofcompact and noncompact type.In both cases, we have two classes of symmetric spacesdescribed in terms of Riemannian symmetric pairs (G ,K ).
35 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Rank and Sectional Curvature
The rank of is at least one with equality if the sectionalcurvature is positive or negative. If the sectional curvature ispositive the space is of compact type, and if the sectionalcurvature is negative the space is of noncompact type.The rank of a Euclidean type space is equal to its dimension.This implies Euclidean type spaces are isometric to Euclideanspace of that dimension.Therefore we are reduces to classifying symmetric spaces ofcompact and noncompact type.In both cases, we have two classes of symmetric spacesdescribed in terms of Riemannian symmetric pairs (G ,K ).
35 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Rank and Sectional Curvature
The rank of is at least one with equality if the sectionalcurvature is positive or negative. If the sectional curvature ispositive the space is of compact type, and if the sectionalcurvature is negative the space is of noncompact type.The rank of a Euclidean type space is equal to its dimension.This implies Euclidean type spaces are isometric to Euclideanspace of that dimension.Therefore we are reduces to classifying symmetric spaces ofcompact and noncompact type.In both cases, we have two classes of symmetric spacesdescribed in terms of Riemannian symmetric pairs (G ,K ).
35 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Rank and Sectional Curvature
The rank of is at least one with equality if the sectionalcurvature is positive or negative. If the sectional curvature ispositive the space is of compact type, and if the sectionalcurvature is negative the space is of noncompact type.The rank of a Euclidean type space is equal to its dimension.This implies Euclidean type spaces are isometric to Euclideanspace of that dimension.Therefore we are reduces to classifying symmetric spaces ofcompact and noncompact type.In both cases, we have two classes of symmetric spacesdescribed in terms of Riemannian symmetric pairs (G ,K ).
35 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Rank and Sectional Curvature
The rank of is at least one with equality if the sectionalcurvature is positive or negative. If the sectional curvature ispositive the space is of compact type, and if the sectionalcurvature is negative the space is of noncompact type.The rank of a Euclidean type space is equal to its dimension.This implies Euclidean type spaces are isometric to Euclideanspace of that dimension.Therefore we are reduces to classifying symmetric spaces ofcompact and noncompact type.In both cases, we have two classes of symmetric spacesdescribed in terms of Riemannian symmetric pairs (G ,K ).
35 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Rank and Sectional Curvature
The rank of is at least one with equality if the sectionalcurvature is positive or negative. If the sectional curvature ispositive the space is of compact type, and if the sectionalcurvature is negative the space is of noncompact type.The rank of a Euclidean type space is equal to its dimension.This implies Euclidean type spaces are isometric to Euclideanspace of that dimension.Therefore we are reduces to classifying symmetric spaces ofcompact and noncompact type.In both cases, we have two classes of symmetric spacesdescribed in terms of Riemannian symmetric pairs (G ,K ).
35 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Classification
Compact type:
• G = H × H where H is a simply connected compact Liegroup and K is the diagonal subgroup.
• G is the complexification of a simply connectednoncompact simple Lie group and K is the maximalcompact subgroup.
Noncompact type:
• G is a simply connected complex simple Lie group and Kis the maximal compact subgroup.
• G is a simply connected noncompact simple Lie groupand K is the maximal compact subgroup.
By the classification of Lie groups all such symmetric spacesare also classified, and this finishes Cartan’s classification.
36 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Classification
Compact type:
• G = H × H where H is a simply connected compact Liegroup and K is the diagonal subgroup.
• G is the complexification of a simply connectednoncompact simple Lie group and K is the maximalcompact subgroup.
Noncompact type:
• G is a simply connected complex simple Lie group and Kis the maximal compact subgroup.
• G is a simply connected noncompact simple Lie groupand K is the maximal compact subgroup.
By the classification of Lie groups all such symmetric spacesare also classified, and this finishes Cartan’s classification.
36 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Classification
Compact type:
• G = H × H where H is a simply connected compact Liegroup and K is the diagonal subgroup.
• G is the complexification of a simply connectednoncompact simple Lie group and K is the maximalcompact subgroup.
Noncompact type:
• G is a simply connected complex simple Lie group and Kis the maximal compact subgroup.
• G is a simply connected noncompact simple Lie groupand K is the maximal compact subgroup.
By the classification of Lie groups all such symmetric spacesare also classified, and this finishes Cartan’s classification.
36 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
Outline
1 History
2 Symmetric Spaces
3 Examples of Symmetric Spaces
4 Curvature and Locally Symmetric Spaces
5 Effective Orthogonal Symmetric Lie Algebras and the KillingForm
6 Decomposition of Symmetric Spaces
7 Rank and Classification
8 References
37 / 38
Symmetricspaces and
Cartan’sclassification
Henry Twiss
History
SymmetricSpaces
Examples ofSymmetricSpaces
Curvature andLocallySymmetricSpaces
EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form
Decompositionof SymmetricSpaces
Rank andClassification
References
References
Thanks!
• J. Eschenburg: Lecture Notes on Symmetric Spaces,University of Augsburg.
• X. Gao: Symmetric Spaces, University of Illinois, 2014.
• P. Holmelin: Symmetric Spaces, Master’s Thesis, LundInstitute of Technology, 2005.
• P. Petersen: Riemannian Geometry Third Edition,Springer, AG, Switzerland, 2016.
38 / 38