pseudo-riemannian manifolds and isometric …gelosp2013/files/quiroga.pdfactions of simple lie...

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

Pseudo-Riemannian Manifolds and IsometricActions of Simple Lie Groups

Raul Quiroga-Barranco

CIMAT, Guanajuato, Mexico

7th International Meeting on Lorentzian Geometry, 2013Sao Paulo, Brazil

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

1 Geometric actions of simple groups

2 Rigidity or structure results

3 Geometric actions and their symmetries

4 Killing fields fixing points

1

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

Let H be a connected noncompact simple Lie group acting smoothly on amanifold M.

Also, assume that H preserves a geometric structure on M.Some examples:

If K is a maximal compact subgroup of H, then H acts by isometrieson the Riemannian symmetric space H/K .

If P is a parabolic subgroup of H, then H acts on H/P preserving aparabolic geometry.

If Γ is a lattice in H, i.e. a discrete subgroup with finite covolume inH, then H acts on H/Γ preserving a pseudo-Riemannian metric offinite volume.

Our focus will be in the last example.

2

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

Let H be a connected noncompact simple Lie group acting smoothly on amanifold M.Also, assume that H preserves a geometric structure on M.

Some examples:

If K is a maximal compact subgroup of H, then H acts by isometrieson the Riemannian symmetric space H/K .

If P is a parabolic subgroup of H, then H acts on H/P preserving aparabolic geometry.

If Γ is a lattice in H, i.e. a discrete subgroup with finite covolume inH, then H acts on H/Γ preserving a pseudo-Riemannian metric offinite volume.

Our focus will be in the last example.

2

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

Let H be a connected noncompact simple Lie group acting smoothly on amanifold M.Also, assume that H preserves a geometric structure on M.Some examples:

If K is a maximal compact subgroup of H, then H acts by isometrieson the Riemannian symmetric space H/K .

If P is a parabolic subgroup of H, then H acts on H/P preserving aparabolic geometry.

If Γ is a lattice in H, i.e. a discrete subgroup with finite covolume inH, then H acts on H/Γ preserving a pseudo-Riemannian metric offinite volume.

Our focus will be in the last example.

2

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

Let H be a connected noncompact simple Lie group acting smoothly on amanifold M.Also, assume that H preserves a geometric structure on M.Some examples:

If K is a maximal compact subgroup of H, then H acts by isometrieson the Riemannian symmetric space H/K .

If P is a parabolic subgroup of H, then H acts on H/P preserving aparabolic geometry.

If Γ is a lattice in H, i.e. a discrete subgroup with finite covolume inH, then H acts on H/Γ preserving a pseudo-Riemannian metric offinite volume.

Our focus will be in the last example.

2

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

Let H be a connected noncompact simple Lie group acting smoothly on amanifold M.Also, assume that H preserves a geometric structure on M.Some examples:

If K is a maximal compact subgroup of H, then H acts by isometrieson the Riemannian symmetric space H/K .

If P is a parabolic subgroup of H, then H acts on H/P preserving aparabolic geometry.

If Γ is a lattice in H, i.e. a discrete subgroup with finite covolume inH, then H acts on H/Γ preserving a pseudo-Riemannian metric offinite volume.

Our focus will be in the last example.

2

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

Let H be a connected noncompact simple Lie group acting smoothly on amanifold M.Also, assume that H preserves a geometric structure on M.Some examples:

If K is a maximal compact subgroup of H, then H acts by isometrieson the Riemannian symmetric space H/K .

If P is a parabolic subgroup of H, then H acts on H/P preserving aparabolic geometry.

If Γ is a lattice in H, i.e. a discrete subgroup with finite covolume inH, then H acts on H/Γ preserving a pseudo-Riemannian metric offinite volume.

Our focus will be in the last example.

2

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

Let H be a connected noncompact simple Lie group acting smoothly on amanifold M.Also, assume that H preserves a geometric structure on M.Some examples:

If K is a maximal compact subgroup of H, then H acts by isometrieson the Riemannian symmetric space H/K .

If P is a parabolic subgroup of H, then H acts on H/P preserving aparabolic geometry.

If Γ is a lattice in H, i.e. a discrete subgroup with finite covolume inH, then H acts on H/Γ preserving a pseudo-Riemannian metric offinite volume.

Our focus will be in the last example.

2

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

Problem: To what extent can the pseudo-Riemannian manifold H/Γ bedetermined by its isometries?

The group H of left translations trivially determines H/Γ, so we look forsmaller groups of isometries.

Model example

If G is a connected noncompact simple subgroup of H, then G acts onH/Γ preserving a finite volume pseudo-Riemannian metric.

We can further modify this model as follows.

Model example

Let K be a compact subgroup of H with K ∩ Γ = e, so that K\H/Γ is afinite volume pseudo-Riemannian manifold. If G and K centralize eachother, then G acts by isometries on the double coset space K\H/Γ.

Problem: Does the G -action above determine the pseudo-Riemannianmanifold H/Γ?

3

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

Problem: To what extent can the pseudo-Riemannian manifold H/Γ bedetermined by its isometries?The group H of left translations trivially determines H/Γ, so we look forsmaller groups of isometries.

Model example

If G is a connected noncompact simple subgroup of H, then G acts onH/Γ preserving a finite volume pseudo-Riemannian metric.

We can further modify this model as follows.

Model example

Let K be a compact subgroup of H with K ∩ Γ = e, so that K\H/Γ is afinite volume pseudo-Riemannian manifold. If G and K centralize eachother, then G acts by isometries on the double coset space K\H/Γ.

Problem: Does the G -action above determine the pseudo-Riemannianmanifold H/Γ?

3

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

Problem: To what extent can the pseudo-Riemannian manifold H/Γ bedetermined by its isometries?The group H of left translations trivially determines H/Γ, so we look forsmaller groups of isometries.

Model example

If G is a connected noncompact simple subgroup of H, then G acts onH/Γ preserving a finite volume pseudo-Riemannian metric.

We can further modify this model as follows.

Model example

Let K be a compact subgroup of H with K ∩ Γ = e, so that K\H/Γ is afinite volume pseudo-Riemannian manifold. If G and K centralize eachother, then G acts by isometries on the double coset space K\H/Γ.

Problem: Does the G -action above determine the pseudo-Riemannianmanifold H/Γ?

3

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

Problem: To what extent can the pseudo-Riemannian manifold H/Γ bedetermined by its isometries?The group H of left translations trivially determines H/Γ, so we look forsmaller groups of isometries.

Model example

If G is a connected noncompact simple subgroup of H, then G acts onH/Γ preserving a finite volume pseudo-Riemannian metric.

We can further modify this model as follows.

Model example

Let K be a compact subgroup of H with K ∩ Γ = e, so that K\H/Γ is afinite volume pseudo-Riemannian manifold. If G and K centralize eachother, then G acts by isometries on the double coset space K\H/Γ.

Problem: Does the G -action above determine the pseudo-Riemannianmanifold H/Γ?

3

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

Problem: To what extent can the pseudo-Riemannian manifold H/Γ bedetermined by its isometries?The group H of left translations trivially determines H/Γ, so we look forsmaller groups of isometries.

Model example

If G is a connected noncompact simple subgroup of H, then G acts onH/Γ preserving a finite volume pseudo-Riemannian metric.

We can further modify this model as follows.

Model example

Let K be a compact subgroup of H with K ∩ Γ = e, so that K\H/Γ is afinite volume pseudo-Riemannian manifold.

If G and K centralize eachother, then G acts by isometries on the double coset space K\H/Γ.

Problem: Does the G -action above determine the pseudo-Riemannianmanifold H/Γ?

3

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

Problem: To what extent can the pseudo-Riemannian manifold H/Γ bedetermined by its isometries?The group H of left translations trivially determines H/Γ, so we look forsmaller groups of isometries.

Model example

If G is a connected noncompact simple subgroup of H, then G acts onH/Γ preserving a finite volume pseudo-Riemannian metric.

We can further modify this model as follows.

Model example

Let K be a compact subgroup of H with K ∩ Γ = e, so that K\H/Γ is afinite volume pseudo-Riemannian manifold. If G and K centralize eachother, then G acts by isometries on the double coset space K\H/Γ.

Problem: Does the G -action above determine the pseudo-Riemannianmanifold H/Γ?

3

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

Problem: To what extent can the pseudo-Riemannian manifold H/Γ bedetermined by its isometries?The group H of left translations trivially determines H/Γ, so we look forsmaller groups of isometries.

Model example

If G is a connected noncompact simple subgroup of H, then G acts onH/Γ preserving a finite volume pseudo-Riemannian metric.

We can further modify this model as follows.

Model example

Let K be a compact subgroup of H with K ∩ Γ = e, so that K\H/Γ is afinite volume pseudo-Riemannian manifold. If G and K centralize eachother, then G acts by isometries on the double coset space K\H/Γ.

Problem: Does the G -action above determine the pseudo-Riemannianmanifold H/Γ?

3

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

From now on, we consider the following setup.

A smooth manifold M carrying a pseudo-Riemannian metric of finitevolume.

An isometric G -action on M.

And assume that G -action on M has a dense orbit.

It has been conjectured that the double coset examples are the onlypossibility for M.

Zimmer’s program

Prove that every G -action on M as above can always “essentially” beobtained by a double coset space K\H/Γ.

4

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

From now on, we consider the following setup.

A smooth manifold M carrying a pseudo-Riemannian metric of finitevolume.

An isometric G -action on M.

And assume that G -action on M has a dense orbit.

It has been conjectured that the double coset examples are the onlypossibility for M.

Zimmer’s program

Prove that every G -action on M as above can always “essentially” beobtained by a double coset space K\H/Γ.

4

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

From now on, we consider the following setup.

A smooth manifold M carrying a pseudo-Riemannian metric of finitevolume.

An isometric G -action on M.

And assume that G -action on M has a dense orbit.

It has been conjectured that the double coset examples are the onlypossibility for M.

Zimmer’s program

Prove that every G -action on M as above can always “essentially” beobtained by a double coset space K\H/Γ.

4

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

From now on, we consider the following setup.

A smooth manifold M carrying a pseudo-Riemannian metric of finitevolume.

An isometric G -action on M.

And assume that G -action on M has a dense orbit.

It has been conjectured that the double coset examples are the onlypossibility for M.

Zimmer’s program

Prove that every G -action on M as above can always “essentially” beobtained by a double coset space K\H/Γ.

4

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

From now on, we consider the following setup.

A smooth manifold M carrying a pseudo-Riemannian metric of finitevolume.

An isometric G -action on M.

And assume that G -action on M has a dense orbit.

It has been conjectured that the double coset examples are the onlypossibility for M.

Zimmer’s program

Prove that every G -action on M as above can always “essentially” beobtained by a double coset space K\H/Γ.

4

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

From now on, we consider the following setup.

A smooth manifold M carrying a pseudo-Riemannian metric of finitevolume.

An isometric G -action on M.

And assume that G -action on M has a dense orbit.

It has been conjectured that the double coset examples are the onlypossibility for M.

Zimmer’s program

Prove that every G -action on M as above can always “essentially” beobtained by a double coset space K\H/Γ.

4

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

1 Geometric actions of simple groups

2 Rigidity or structure results

3 Geometric actions and their symmetries

4 Killing fields fixing points

5

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

For a pseudo-Riemannian manifold, its nullity is the dimension of themaximal null tangent subspaces.

Theorem (Gromov)

Let G be a connected noncompact simple Lie group acting isometricallyon a finite volume pseudo-Riemannian manifold M and with a dense orbit.Then:

nullity of G ≤ nullity of M.

Theorem (Q)

If the equality holds above and either M is compact or complete, thenthere exist

a semisimple Lie group H,

a compact subgroup K of H, and

an irreducible lattice Γ of G × H,

such that M ' (G × K\H)/Γ up to a finite covering map as G -spaces.

6

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

For a pseudo-Riemannian manifold, its nullity is the dimension of themaximal null tangent subspaces.

Theorem (Gromov)

Let G be a connected noncompact simple Lie group acting isometricallyon a finite volume pseudo-Riemannian manifold M and with a dense orbit.Then:

nullity of G ≤ nullity of M.

Theorem (Q)

If the equality holds above and either M is compact or complete, thenthere exist

a semisimple Lie group H,

a compact subgroup K of H, and

an irreducible lattice Γ of G × H,

such that M ' (G × K\H)/Γ up to a finite covering map as G -spaces.

6

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

For a pseudo-Riemannian manifold, its nullity is the dimension of themaximal null tangent subspaces.

Theorem (Gromov)

Let G be a connected noncompact simple Lie group acting isometricallyon a finite volume pseudo-Riemannian manifold M and with a dense orbit.Then:

nullity of G ≤ nullity of M.

Theorem (Q)

If the equality holds above and either M is compact or complete, thenthere exist

a semisimple Lie group H,

a compact subgroup K of H, and

an irreducible lattice Γ of G × H,

such that M ' (G × K\H)/Γ up to a finite covering map as G -spaces.

6

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

For a pseudo-Riemannian manifold, its nullity is the dimension of themaximal null tangent subspaces.

Theorem (Gromov)

Let G be a connected noncompact simple Lie group acting isometricallyon a finite volume pseudo-Riemannian manifold M and with a dense orbit.Then:

nullity of G ≤ nullity of M.

Theorem (Q)

If the equality holds above and either M is compact or complete, thenthere exist

a semisimple Lie group H,

a compact subgroup K of H, and

an irreducible lattice Γ of G × H,

such that M ' (G × K\H)/Γ up to a finite covering map as G -spaces.

6

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

For a pseudo-Riemannian manifold, its nullity is the dimension of themaximal null tangent subspaces.

Theorem (Gromov)

Let G be a connected noncompact simple Lie group acting isometricallyon a finite volume pseudo-Riemannian manifold M and with a dense orbit.Then:

nullity of G ≤ nullity of M.

Theorem (Q)

If the equality holds above and either M is compact or complete, thenthere exist

a semisimple Lie group H,

a compact subgroup K of H, and

an irreducible lattice Γ of G × H,

such that M ' (G × K\H)/Γ up to a finite covering map as G -spaces.

6

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

For a pseudo-Riemannian manifold, its nullity is the dimension of themaximal null tangent subspaces.

Theorem (Gromov)

Let G be a connected noncompact simple Lie group acting isometricallyon a finite volume pseudo-Riemannian manifold M and with a dense orbit.Then:

nullity of G ≤ nullity of M.

Theorem (Q)

If the equality holds above and either M is compact or complete, thenthere exist

a semisimple Lie group H,

a compact subgroup K of H, and

an irreducible lattice Γ of G × H,

such that M ' (G × K\H)/Γ up to a finite covering map as G -spaces.

6

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

For a pseudo-Riemannian manifold, its nullity is the dimension of themaximal null tangent subspaces.

Theorem (Gromov)

Let G be a connected noncompact simple Lie group acting isometricallyon a finite volume pseudo-Riemannian manifold M and with a dense orbit.Then:

nullity of G ≤ nullity of M.

Theorem (Q)

If the equality holds above and either M is compact or complete, thenthere exist

a semisimple Lie group H,

a compact subgroup K of H, and

an irreducible lattice Γ of G × H,

such that M ' (G × K\H)/Γ up to a finite covering map as G -spaces.

6

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

The following result holds for actions of the pseudo-orthogonal group onlow dimensional manifolds.

Theorem (Olafsson-Q)

Let p, q ≥ 1, n = p + q ≥ 5, and suppose that SO0(p, q) actsisometrically with a dense orbit on a connected complete analytic finitevolume pseudo-Riemannian manifold M.If we have

dimM ≤ dim SO0(p, q) + n.

then, either one of the following holds.

The universal covering of M isometrically splits M = SO0(p, q)× N.

For H either SO0(p + 1, q) or SO0(p, q + 1), there is a lattice Γ ofH such that M ' H/Γ up to a finite covering map as G -spaces.

7

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

The following result holds for actions of the pseudo-orthogonal group onlow dimensional manifolds.

Theorem (Olafsson-Q)

Let p, q ≥ 1, n = p + q ≥ 5,

and suppose that SO0(p, q) actsisometrically with a dense orbit on a connected complete analytic finitevolume pseudo-Riemannian manifold M.If we have

dimM ≤ dim SO0(p, q) + n.

then, either one of the following holds.

The universal covering of M isometrically splits M = SO0(p, q)× N.

For H either SO0(p + 1, q) or SO0(p, q + 1), there is a lattice Γ ofH such that M ' H/Γ up to a finite covering map as G -spaces.

7

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

The following result holds for actions of the pseudo-orthogonal group onlow dimensional manifolds.

Theorem (Olafsson-Q)

Let p, q ≥ 1, n = p + q ≥ 5, and suppose that SO0(p, q) actsisometrically with a dense orbit on a connected complete analytic finitevolume pseudo-Riemannian manifold M.

If we havedimM ≤ dim SO0(p, q) + n.

then, either one of the following holds.

The universal covering of M isometrically splits M = SO0(p, q)× N.

For H either SO0(p + 1, q) or SO0(p, q + 1), there is a lattice Γ ofH such that M ' H/Γ up to a finite covering map as G -spaces.

7

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

The following result holds for actions of the pseudo-orthogonal group onlow dimensional manifolds.

Theorem (Olafsson-Q)

Let p, q ≥ 1, n = p + q ≥ 5, and suppose that SO0(p, q) actsisometrically with a dense orbit on a connected complete analytic finitevolume pseudo-Riemannian manifold M.If we have

dimM ≤ dim SO0(p, q) + n.

then, either one of the following holds.

The universal covering of M isometrically splits M = SO0(p, q)× N.

For H either SO0(p + 1, q) or SO0(p, q + 1), there is a lattice Γ ofH such that M ' H/Γ up to a finite covering map as G -spaces.

7

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

The following result holds for actions of the pseudo-orthogonal group onlow dimensional manifolds.

Theorem (Olafsson-Q)

Let p, q ≥ 1, n = p + q ≥ 5, and suppose that SO0(p, q) actsisometrically with a dense orbit on a connected complete analytic finitevolume pseudo-Riemannian manifold M.If we have

dimM ≤ dim SO0(p, q) + n.

then, either one of the following holds.

The universal covering of M isometrically splits M = SO0(p, q)× N.

For H either SO0(p + 1, q) or SO0(p, q + 1), there is a lattice Γ ofH such that M ' H/Γ up to a finite covering map as G -spaces.

7

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

The following result holds for actions of the pseudo-orthogonal group onlow dimensional manifolds.

Theorem (Olafsson-Q)

Let p, q ≥ 1, n = p + q ≥ 5, and suppose that SO0(p, q) actsisometrically with a dense orbit on a connected complete analytic finitevolume pseudo-Riemannian manifold M.If we have

dimM ≤ dim SO0(p, q) + n.

then, either one of the following holds.

The universal covering of M isometrically splits M = SO0(p, q)× N.

For H either SO0(p + 1, q) or SO0(p, q + 1), there is a lattice Γ ofH such that M ' H/Γ up to a finite covering map as G -spaces.

7

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

The following result holds for actions of the pseudo-orthogonal group onlow dimensional manifolds.

Theorem (Olafsson-Q)

Let p, q ≥ 1, n = p + q ≥ 5, and suppose that SO0(p, q) actsisometrically with a dense orbit on a connected complete analytic finitevolume pseudo-Riemannian manifold M.If we have

dimM ≤ dim SO0(p, q) + n.

then, either one of the following holds.

The universal covering of M isometrically splits M = SO0(p, q)× N.

For H either SO0(p + 1, q) or SO0(p, q + 1), there is a lattice Γ ofH such that M ' H/Γ up to a finite covering map as G -spaces.

7

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

There are several other results that provide important information on themanifold acted upon.

Theorem (Candel-Q, Gromov, Zimmer)

If G is a connected noncompact simple Lie group acting ergodically on amanifold M and preserving a unimodular, finite type and algebraicgeometric structure, then there exists a representation

ρ : π1(M)→ GL(n,R)

so that the Zariski closure of ρ(π1(M)) contains a group locallyisomorphic to G .

8

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

There are several other results that provide important information on themanifold acted upon.

Theorem (Candel-Q, Gromov, Zimmer)

If G is a connected noncompact simple Lie group acting ergodically on amanifold M and preserving a unimodular, finite type and algebraicgeometric structure, then there exists a representation

ρ : π1(M)→ GL(n,R)

so that the Zariski closure of ρ(π1(M)) contains a group locallyisomorphic to G .

8

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

1 Geometric actions of simple groups

2 Rigidity or structure results

3 Geometric actions and their symmetries

4 Killing fields fixing points

9

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

For the G -action on H/Γ described before, the group of isometriescontains G .

Furthermore, there are some additional local isometries.Consider the diagram

H

��

Hoo

G //

==

H/Γ

The right H-action does not descend to H/Γ, but it yields:

A local H-action on H/Γ commuting with the G -action, and so

a Lie algebra of local Killing vector fields commuting with theG -action.

10

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

For the G -action on H/Γ described before, the group of isometriescontains G .Furthermore, there are some additional local isometries.

Consider the diagram

H

��

Hoo

G //

==

H/Γ

The right H-action does not descend to H/Γ, but it yields:

A local H-action on H/Γ commuting with the G -action, and so

a Lie algebra of local Killing vector fields commuting with theG -action.

10

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

For the G -action on H/Γ described before, the group of isometriescontains G .Furthermore, there are some additional local isometries.Consider the diagram

H

��

Hoo

G //

==

H/Γ

The right H-action does not descend to H/Γ, but it yields:

A local H-action on H/Γ commuting with the G -action, and so

a Lie algebra of local Killing vector fields commuting with theG -action.

10

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

For the G -action on H/Γ described before, the group of isometriescontains G .Furthermore, there are some additional local isometries.Consider the diagram

H

��

Hoo

G //

==

H/Γ

The right H-action does not descend to H/Γ, but it yields:

A local H-action on H/Γ commuting with the G -action, and so

a Lie algebra of local Killing vector fields commuting with theG -action.

10

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

For the G -action on H/Γ described before, the group of isometriescontains G .Furthermore, there are some additional local isometries.Consider the diagram

H

��

Hoo

G //

==

H/Γ

The right H-action does not descend to H/Γ, but it yields:

A local H-action on H/Γ commuting with the G -action, and so

a Lie algebra of local Killing vector fields commuting with theG -action.

10

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

For the G -action on H/Γ described before, the group of isometriescontains G .Furthermore, there are some additional local isometries.Consider the diagram

H

��

Hoo

G //

==

H/Γ

The right H-action does not descend to H/Γ, but it yields:

A local H-action on H/Γ commuting with the G -action, and so

a Lie algebra of local Killing vector fields commuting with theG -action.

10

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

Since G ⊂ H, we also have conjugation actions fixing points in H.

For every, h0 ∈ H

G × H → H

(g , h) 7→ g(hh−10 )g−1

is an isometric action that fixes h0.This action does not descend to H/Γ, but it yields:

A local G -action on H/Γ fixing a given point, and so

a Lie algebra of local Killing fields vanishing at a given point.

Remark

The model example of the isometric G -action on H/Γ comes withadditional local symmetries given by local Killing fields.

11

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

Since G ⊂ H, we also have conjugation actions fixing points in H.For every, h0 ∈ H

G × H → H

(g , h) 7→ g(hh−10 )g−1

is an isometric action that fixes h0.

This action does not descend to H/Γ, but it yields:

A local G -action on H/Γ fixing a given point, and so

a Lie algebra of local Killing fields vanishing at a given point.

Remark

The model example of the isometric G -action on H/Γ comes withadditional local symmetries given by local Killing fields.

11

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

Since G ⊂ H, we also have conjugation actions fixing points in H.For every, h0 ∈ H

G × H → H

(g , h) 7→ g(hh−10 )g−1

is an isometric action that fixes h0.This action does not descend to H/Γ, but it yields:

A local G -action on H/Γ fixing a given point, and so

a Lie algebra of local Killing fields vanishing at a given point.

Remark

The model example of the isometric G -action on H/Γ comes withadditional local symmetries given by local Killing fields.

11

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

Since G ⊂ H, we also have conjugation actions fixing points in H.For every, h0 ∈ H

G × H → H

(g , h) 7→ g(hh−10 )g−1

is an isometric action that fixes h0.This action does not descend to H/Γ, but it yields:

A local G -action on H/Γ fixing a given point, and so

a Lie algebra of local Killing fields vanishing at a given point.

Remark

The model example of the isometric G -action on H/Γ comes withadditional local symmetries given by local Killing fields.

11

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

Since G ⊂ H, we also have conjugation actions fixing points in H.For every, h0 ∈ H

G × H → H

(g , h) 7→ g(hh−10 )g−1

is an isometric action that fixes h0.This action does not descend to H/Γ, but it yields:

A local G -action on H/Γ fixing a given point, and so

a Lie algebra of local Killing fields vanishing at a given point.

Remark

The model example of the isometric G -action on H/Γ comes withadditional local symmetries given by local Killing fields.

11

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

Since G ⊂ H, we also have conjugation actions fixing points in H.For every, h0 ∈ H

G × H → H

(g , h) 7→ g(hh−10 )g−1

is an isometric action that fixes h0.This action does not descend to H/Γ, but it yields:

A local G -action on H/Γ fixing a given point, and so

a Lie algebra of local Killing fields vanishing at a given point.

Remark

The model example of the isometric G -action on H/Γ comes withadditional local symmetries given by local Killing fields.

11

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

1 Geometric actions of simple groups

2 Rigidity or structure results

3 Geometric actions and their symmetries

4 Killing fields fixing points

12

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

It is a remarkable fact that there exist symmetries fixing points on themanifolds M with a G -action as before in the general case.

Theorem (Gromov, Q, Zimmer)

Let M be a manifold with a G -action satisfying the geometric hypothesisdescribed above. Then, for a.e. x ∈ M there is a Lie algebra g(x) of localKilling fields defined in a neighborhood of x satisfying the followingproperties.

Every element of g(x) vanishes at x , i.e. the correspondingisometries fix x .

We have g(x) ' g = Lie(G ).

The (local) isometries defined by g(x) preserve the G -orbits.

In other words, g(x) is a Lie algebra of symmetries realizing a conjugationon M at the point x .

13

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

It is a remarkable fact that there exist symmetries fixing points on themanifolds M with a G -action as before in the general case.

Theorem (Gromov, Q, Zimmer)

Let M be a manifold with a G -action satisfying the geometric hypothesisdescribed above.

Then, for a.e. x ∈ M there is a Lie algebra g(x) of localKilling fields defined in a neighborhood of x satisfying the followingproperties.

Every element of g(x) vanishes at x , i.e. the correspondingisometries fix x .

We have g(x) ' g = Lie(G ).

The (local) isometries defined by g(x) preserve the G -orbits.

In other words, g(x) is a Lie algebra of symmetries realizing a conjugationon M at the point x .

13

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

It is a remarkable fact that there exist symmetries fixing points on themanifolds M with a G -action as before in the general case.

Theorem (Gromov, Q, Zimmer)

Let M be a manifold with a G -action satisfying the geometric hypothesisdescribed above. Then, for a.e. x ∈ M there is a Lie algebra g(x) of localKilling fields defined in a neighborhood of x satisfying the followingproperties.

Every element of g(x) vanishes at x , i.e. the correspondingisometries fix x .

We have g(x) ' g = Lie(G ).

The (local) isometries defined by g(x) preserve the G -orbits.

In other words, g(x) is a Lie algebra of symmetries realizing a conjugationon M at the point x .

13

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

It is a remarkable fact that there exist symmetries fixing points on themanifolds M with a G -action as before in the general case.

Theorem (Gromov, Q, Zimmer)

Let M be a manifold with a G -action satisfying the geometric hypothesisdescribed above. Then, for a.e. x ∈ M there is a Lie algebra g(x) of localKilling fields defined in a neighborhood of x satisfying the followingproperties.

Every element of g(x) vanishes at x , i.e. the correspondingisometries fix x .

We have g(x) ' g = Lie(G ).

The (local) isometries defined by g(x) preserve the G -orbits.

In other words, g(x) is a Lie algebra of symmetries realizing a conjugationon M at the point x .

13

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

It is a remarkable fact that there exist symmetries fixing points on themanifolds M with a G -action as before in the general case.

Theorem (Gromov, Q, Zimmer)

Let M be a manifold with a G -action satisfying the geometric hypothesisdescribed above. Then, for a.e. x ∈ M there is a Lie algebra g(x) of localKilling fields defined in a neighborhood of x satisfying the followingproperties.

Every element of g(x) vanishes at x , i.e. the correspondingisometries fix x .

We have g(x) ' g = Lie(G ).

The (local) isometries defined by g(x) preserve the G -orbits.

In other words, g(x) is a Lie algebra of symmetries realizing a conjugationon M at the point x .

13

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

It is a remarkable fact that there exist symmetries fixing points on themanifolds M with a G -action as before in the general case.

Theorem (Gromov, Q, Zimmer)

Let M be a manifold with a G -action satisfying the geometric hypothesisdescribed above. Then, for a.e. x ∈ M there is a Lie algebra g(x) of localKilling fields defined in a neighborhood of x satisfying the followingproperties.

Every element of g(x) vanishes at x , i.e. the correspondingisometries fix x .

We have g(x) ' g = Lie(G ).

The (local) isometries defined by g(x) preserve the G -orbits.

In other words, g(x) is a Lie algebra of symmetries realizing a conjugationon M at the point x .

13

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

It is a remarkable fact that there exist symmetries fixing points on themanifolds M with a G -action as before in the general case.

Theorem (Gromov, Q, Zimmer)

Let M be a manifold with a G -action satisfying the geometric hypothesisdescribed above. Then, for a.e. x ∈ M there is a Lie algebra g(x) of localKilling fields defined in a neighborhood of x satisfying the followingproperties.

Every element of g(x) vanishes at x , i.e. the correspondingisometries fix x .

We have g(x) ' g = Lie(G ).

The (local) isometries defined by g(x) preserve the G -orbits.

In other words, g(x) is a Lie algebra of symmetries realizing a conjugationon M at the point x .

13

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

Next, to obtain a right action we just add the left action and theconjugation.

For X ∈ g, we will denote with X ∗ the Killing field on M generated bythe one-parameter group of isometries (exp(tX ))t∈R ⊂ G acting on M.

Theorem (Gromov, Q, Zimmer)

For the above setup, for a.e. x ∈ M, there is an isomorphismρx : g→ g(x), such that the space:

G(x) = {X ∗ + ρx(X ) : X ∈ g}

satisfies the following properties.

G(x) is a Lie algebra isomorphic to g.

G(x) consists of Killing fields defined in a neighborhood of x in M.

The G -action commutes with every element in G(x).

In other words, G(x) is a Lie algebra of symmetries realizing a rightaction on M at the point x .

14

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

Next, to obtain a right action we just add the left action and theconjugation.For X ∈ g, we will denote with X ∗ the Killing field on M generated bythe one-parameter group of isometries (exp(tX ))t∈R ⊂ G acting on M.

Theorem (Gromov, Q, Zimmer)

For the above setup, for a.e. x ∈ M, there is an isomorphismρx : g→ g(x), such that the space:

G(x) = {X ∗ + ρx(X ) : X ∈ g}

satisfies the following properties.

G(x) is a Lie algebra isomorphic to g.

G(x) consists of Killing fields defined in a neighborhood of x in M.

The G -action commutes with every element in G(x).

In other words, G(x) is a Lie algebra of symmetries realizing a rightaction on M at the point x .

14

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

Next, to obtain a right action we just add the left action and theconjugation.For X ∈ g, we will denote with X ∗ the Killing field on M generated bythe one-parameter group of isometries (exp(tX ))t∈R ⊂ G acting on M.

Theorem (Gromov, Q, Zimmer)

For the above setup, for a.e. x ∈ M, there is an isomorphismρx : g→ g(x), such that the space:

G(x) = {X ∗ + ρx(X ) : X ∈ g}

satisfies the following properties.

G(x) is a Lie algebra isomorphic to g.

G(x) consists of Killing fields defined in a neighborhood of x in M.

The G -action commutes with every element in G(x).

In other words, G(x) is a Lie algebra of symmetries realizing a rightaction on M at the point x .

14

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

Next, to obtain a right action we just add the left action and theconjugation.For X ∈ g, we will denote with X ∗ the Killing field on M generated bythe one-parameter group of isometries (exp(tX ))t∈R ⊂ G acting on M.

Theorem (Gromov, Q, Zimmer)

For the above setup, for a.e. x ∈ M, there is an isomorphismρx : g→ g(x), such that the space:

G(x) = {X ∗ + ρx(X ) : X ∈ g}

satisfies the following properties.

G(x) is a Lie algebra isomorphic to g.

G(x) consists of Killing fields defined in a neighborhood of x in M.

The G -action commutes with every element in G(x).

In other words, G(x) is a Lie algebra of symmetries realizing a rightaction on M at the point x .

14

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

Next, to obtain a right action we just add the left action and theconjugation.For X ∈ g, we will denote with X ∗ the Killing field on M generated bythe one-parameter group of isometries (exp(tX ))t∈R ⊂ G acting on M.

Theorem (Gromov, Q, Zimmer)

For the above setup, for a.e. x ∈ M, there is an isomorphismρx : g→ g(x), such that the space:

G(x) = {X ∗ + ρx(X ) : X ∈ g}

satisfies the following properties.

G(x) is a Lie algebra isomorphic to g.

G(x) consists of Killing fields defined in a neighborhood of x in M.

The G -action commutes with every element in G(x).

In other words, G(x) is a Lie algebra of symmetries realizing a rightaction on M at the point x .

14

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

Next, to obtain a right action we just add the left action and theconjugation.For X ∈ g, we will denote with X ∗ the Killing field on M generated bythe one-parameter group of isometries (exp(tX ))t∈R ⊂ G acting on M.

Theorem (Gromov, Q, Zimmer)

For the above setup, for a.e. x ∈ M, there is an isomorphismρx : g→ g(x), such that the space:

G(x) = {X ∗ + ρx(X ) : X ∈ g}

satisfies the following properties.

G(x) is a Lie algebra isomorphic to g.

G(x) consists of Killing fields defined in a neighborhood of x in M.

The G -action commutes with every element in G(x).

In other words, G(x) is a Lie algebra of symmetries realizing a rightaction on M at the point x .

14

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

Next, to obtain a right action we just add the left action and theconjugation.For X ∈ g, we will denote with X ∗ the Killing field on M generated bythe one-parameter group of isometries (exp(tX ))t∈R ⊂ G acting on M.

Theorem (Gromov, Q, Zimmer)

For the above setup, for a.e. x ∈ M, there is an isomorphismρx : g→ g(x), such that the space:

G(x) = {X ∗ + ρx(X ) : X ∈ g}

satisfies the following properties.

G(x) is a Lie algebra isomorphic to g.

G(x) consists of Killing fields defined in a neighborhood of x in M.

The G -action commutes with every element in G(x).

In other words, G(x) is a Lie algebra of symmetries realizing a rightaction on M at the point x .

14

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

From now on, we will assume that the G -invariant geometric structure isa finite volume pseudo-Riemannian metric.

Let O be the foliation on M defined by the G -orbits.In some important cases, we can assume that the leaves of O arenondegenerate for the metric, and we can decompose at every x :

TxM = TxO ⊕ TxO⊥.

Two remarkable facts thus appear.

The structure of TxO is known since it comes from that of G .

Since g(x) preserves the G -orbits and acts by isometries, it preservesTxO⊥. Hence, TxO⊥ is a g-module and we can use representationtheory to complete the description of TxM.

This scheme turns out to be successful to obtain structure results.

15

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

From now on, we will assume that the G -invariant geometric structure isa finite volume pseudo-Riemannian metric.Let O be the foliation on M defined by the G -orbits.

In some important cases, we can assume that the leaves of O arenondegenerate for the metric, and we can decompose at every x :

TxM = TxO ⊕ TxO⊥.

Two remarkable facts thus appear.

The structure of TxO is known since it comes from that of G .

Since g(x) preserves the G -orbits and acts by isometries, it preservesTxO⊥. Hence, TxO⊥ is a g-module and we can use representationtheory to complete the description of TxM.

This scheme turns out to be successful to obtain structure results.

15

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

From now on, we will assume that the G -invariant geometric structure isa finite volume pseudo-Riemannian metric.Let O be the foliation on M defined by the G -orbits.In some important cases, we can assume that the leaves of O arenondegenerate for the metric, and we can decompose at every x :

TxM = TxO ⊕ TxO⊥.

Two remarkable facts thus appear.

The structure of TxO is known since it comes from that of G .

Since g(x) preserves the G -orbits and acts by isometries, it preservesTxO⊥. Hence, TxO⊥ is a g-module and we can use representationtheory to complete the description of TxM.

This scheme turns out to be successful to obtain structure results.

15

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

From now on, we will assume that the G -invariant geometric structure isa finite volume pseudo-Riemannian metric.Let O be the foliation on M defined by the G -orbits.In some important cases, we can assume that the leaves of O arenondegenerate for the metric, and we can decompose at every x :

TxM = TxO ⊕ TxO⊥.

Two remarkable facts thus appear.

The structure of TxO is known since it comes from that of G .

Since g(x) preserves the G -orbits and acts by isometries, it preservesTxO⊥.

Hence, TxO⊥ is a g-module and we can use representationtheory to complete the description of TxM.

This scheme turns out to be successful to obtain structure results.

15

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

From now on, we will assume that the G -invariant geometric structure isa finite volume pseudo-Riemannian metric.Let O be the foliation on M defined by the G -orbits.In some important cases, we can assume that the leaves of O arenondegenerate for the metric, and we can decompose at every x :

TxM = TxO ⊕ TxO⊥.

Two remarkable facts thus appear.

The structure of TxO is known since it comes from that of G .

Since g(x) preserves the G -orbits and acts by isometries, it preservesTxO⊥.

Hence, TxO⊥ is a g-module and we can use representationtheory to complete the description of TxM.

This scheme turns out to be successful to obtain structure results.

15

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

From now on, we will assume that the G -invariant geometric structure isa finite volume pseudo-Riemannian metric.Let O be the foliation on M defined by the G -orbits.In some important cases, we can assume that the leaves of O arenondegenerate for the metric, and we can decompose at every x :

TxM = TxO ⊕ TxO⊥.

Two remarkable facts thus appear.

The structure of TxO is known since it comes from that of G .

Since g(x) preserves the G -orbits and acts by isometries, it preservesTxO⊥.

Hence, TxO⊥ is a g-module and we can use representationtheory to complete the description of TxM.

This scheme turns out to be successful to obtain structure results.

15

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

From now on, we will assume that the G -invariant geometric structure isa finite volume pseudo-Riemannian metric.Let O be the foliation on M defined by the G -orbits.In some important cases, we can assume that the leaves of O arenondegenerate for the metric, and we can decompose at every x :

TxM = TxO ⊕ TxO⊥.

Two remarkable facts thus appear.

The structure of TxO is known since it comes from that of G .

Since g(x) preserves the G -orbits and acts by isometries, it preservesTxO⊥. Hence, TxO⊥ is a g-module and we can use representationtheory to complete the description of TxM.

This scheme turns out to be successful to obtain structure results.

15

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

From now on, we will assume that the G -invariant geometric structure isa finite volume pseudo-Riemannian metric.Let O be the foliation on M defined by the G -orbits.In some important cases, we can assume that the leaves of O arenondegenerate for the metric, and we can decompose at every x :

TxM = TxO ⊕ TxO⊥.

Two remarkable facts thus appear.

The structure of TxO is known since it comes from that of G .

Since g(x) preserves the G -orbits and acts by isometries, it preservesTxO⊥. Hence, TxO⊥ is a g-module and we can use representationtheory to complete the description of TxM.

This scheme turns out to be successful to obtain structure results.

15

Geometric actions of simple groups Rigidity or structure results Geometric actions and their symmetries Killing fields fixing points References

References

A. Candel and R. Quiroga–Barranco, Gromov’s centralizer theorem,Geom. Dedicata 100 (2003), 123–155.

M. Gromov, Rigid transformations groups, in Geometriedifferentielle, Hermann, 1988, 65–139.

G. Olafsson and R. Quiroga-Barranco, On low-dimensional manifoldswith isometric SO0(p, q)-actions. Transform. Groups 17 (2012),no. 3, 835–860.

R. Quiroga-Barranco, Isometric actions of simple Lie groups onpseudo-Riemannian manifolds, Ann. of Math. (2) 164 (2006), no. 6,941–969.

R. J. Zimmer, Automorphism groups and fundamental groups ofgeometric manifolds, 693–710, Proc. Sympos. Pure Math., 54, Part3, Amer. Math. Soc., Providence, RI, 1993.

16