Lagrangian Submanifolds with Constant AngleFunctions in the Nearly Kähler S3 × S3

Burcu BektasIstanbul Technical University, Istanbul, Turkey

Joint work with Marilena Moruz (Université de Valenciennes, France),Joeri Van der Veken (KU Leuven, Belgium) and

Luc Vrancken (Université de Valenciennes, France)

19 Geometrical Seminar, 28 August-4 September 2016

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Basic Notions

An even dimensional manifold is called sympletic manifold if it admits a symplectic form, whichis a closed and nondegenerate 2-form.

Lagrangian Submanifold

A submanifold of a symplectic manifold is called Lagrangian submanifold if the symplectic formrestricted to the manifold vanishes and if the dimension of the submanifold is half the dimension of thesymplectic manifold.

A local classification is trivial from the symplectic point of view. (Darboux Theorem)

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An almost Hermitian manifold (M, J, g) is a manifold M with metric g and almost complexstructure J satisfying

g(JX , JY) = g(X ,Y), X ,Y ∈ TM. (1)

Lagrangian Submanifold

A submanifold in an almost Hermitian manifold is called Lagrangian submanifold if the almost complexstructure interchanges the tangent and the normal spaces and if the dimension is half the dimension.

Lagrangian submanifold in Kähler manifoldB.-Y.Chen and K. Oguie, On totally real submanifolds, Trans. Amer. Math. Soc. 193(1974),257–266.

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A Nearly Kähler Manifold

A nearly Kähler manifold is almost Hermitian manifold (M, g, J) such that ∇J is skew symmetric, i.e.,

(∇X J)Y + (∇Y J)X = 0, X ,Y ∈ TM. (2)

A. Gray, Nearly Kähler manifolds, J. Differential Geometry, 4(1970), 283–309.

P.-A. Nagy, Nearly Kähler geometry and Riemannian folitaions, Asian J. Math., 6(2002),481–504.

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J.-B. Butruille, Homogeneous nearly Kähler manifolds, in: Handbook of Pseudo–RiemannianGeometry and Supersymmetry, 399–423, RMA Lect. Math. Theor. Phys., 16, Eur. Math. Soc. Zürich,2010.The only homogeneous 6–dimensional nearly Kähler manifolds,

the nearly Kähler S6

S3 × S3

the complex projective space CP3

the flag manifold SU(3)/U(1) × U(1)

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The importance of 6–dimensional nearly Kähler manifolds

They serve as building block for arbitrary nearly Kähler manifoldsP.-A. Nagy, Nearly Kähler geometry and Riemannian folitaions, Asian J. Math., 6(2002),481–504.

Six dimensional nearly Kähler manifold is Einstein and there is a bijective correspondencebetween nearly Kähler structure and Killing spinorsT. Friedrich, and R. Grunewald, On the first eigenvalue of the Dirac operator on 6-dimensionalmanifolds, Ann. Global Anal. Geom. 3 (1985), 265–273.

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The nearly Kähler S3 × S3

T(p,q)(S3 × S3) � TpS3 ⊕ TqS

3 (3)

The tangent vector at (p, q): Z(p, q) = (U(p, q),V(p, q))

The almost complex structure J on S3 × S3

JZ(p, q) =1√

3(2pq−1V − U,−2qp−1U + V) (4)

The metric g on S3 × S3

g(Z ,Z ′) =12

(〈Z ,Z ′〉+ 〈JZ , JZ ′〉) (5)

=43

(〈U,U′〉+ 〈V ,V ′〉) −23

(〈p−1U, q−1V ′〉+ 〈p−1U′, q−1V〉) (6)

where 〈·, ·〉 is the product metric, tangent vectors Z = (U,V) and Z ′ = (U′,V ′).

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The almost product structure P on S3 × S3

PZ(p, q) = (pq−1V , qp−1U) (7)

Properties of P

Denote G(X ,Y) = (∇X J)Y .

P2 = Id,PJ = −JP,g(PZ ,PZ ′) = g(Z ,Z ′),PG(X ,Y) + G(PX ,PY) = 0,G(X ,PY) + PG(X ,Y) = −2J(∇X P)Y .

The usual product structure QZ = (−U,V).

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Almost Product Structure on Lagrangian Submanifold

Let M be a Lagrangian submanifold in the nearly Kähler S3 × S3.The pull back of T(S3 × S3) to M splits into TM ⊕ JTM.There are two endomorphisms A ,B : TM −→ TM such that

PX = AX + JBX , ∀X ∈ TM. (8)

A and B are symmetric commuting endomorphisms that satisfy A2 + B2 = I.

The covariant derivative of A and B

(∇X A)Y = BSJX Y − Jh(X ,BY) +12

(JG(X ,AY) − AJG(X ,Y)) (9)

(∇X B)Y = Jh(X ,BY) − ASJX Y +12

(JG(X ,BY) − BJG(X ,Y)). (10)

A and B are symmetric operators whose Lie bracket is zero, i.e., [A ,B] = 0.

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A and B can be diagonalized simultaneously at a point of M and at each point p there is anorthonormal basis e1, e2, e3 ∈ TpM such that

Pei = cos 2θiei + sin 2θiJei , ∀i = 1, 2, 3. (11)

Gauss equation

R(X ,Y)Z =512

(g(Y ,Z)X − g(X ,Z)Y)

+13

(g(AY ,Z)AX − g(AX ,Z)AY + g(BY ,Z)BX − g(BX ,Z)BY)

+ [SJX ,SJY ]Z

(12)

Codazzi equation

(∇h)(X ,Y ,Z) − (∇h)(Y ,X ,Z) =

13

(g(AY ,Z)JBX − g(AX ,Z)JBY − g(BY ,Z)JAX + g(BX ,Z)JAY)

(13)

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For the Levi-Civita connection ∇ on M,

∇Ei Ej = ωkij Ek with ωk

ij = −ωjik

hkij = g(h(Ei ,Ek ), JEk ), hk

ij is totally symmetric.

Lemma (Y. Zhang, Z. Hu, B. Dioos, L. Vrancken, X. Wang, 2016)

Let M be a Lagrangian submanifold. Let {E1,E2,E3} be local orthonormal frame. Denote by hkij and ωk

ijthe components of respectively the second fundamental form and the induced connection. Then wehave

θ1 + θ2 + θ3 is a multiple of π

Ei(θj) = −h ijj

hkij cos(θj − θk ) =

( √3

6 εkij − ω

kij

)sin(θj − θk ), j , k .

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Motivation

Lemma (B. Dioos, L. Vrancken, X. Wang, 2016)

If two of the angles are equal modulo π, then the Lagrangian submanifold is totally geodesic.

Corollary (B. Dioos, L. Vrancken, X. Wang, 2016)

Let M be a Lagrangian submanifold of the nearly Kähler S3 × S3. If M is totally geodesic, then theangles θ1, θ2 and θ3 are constant. Conversely, if the angles are constant and h3

12 = 0, then M is totallygeodesic.

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A. Moroianu and U. Semmelmann, Generalized Killing spinors and Lagrangian graphs, Differ.Geom. Appl. 37(2014), 141–151.

L. Schäfer and K. Smoczyk, Decomposition and minimality of Lagrangian submanifolds innearly Kähler manifolds, Ann. Global Anal. Geom. 37 (2009), no. 3, 221–240.

Theorem A (Y. Zhang, Z. Hu, B. Dioos, L. Vrancken, X. Wang, 2016)

Let M be a totally geodesic Lagrangian submanifold in the nearly Kähler S3 × S3. Then up to anisometry of the nearly Kähler S3 × S3, M is locally congruent with one of the following immersions:

1 f : S3 → S3 × S3 : u 7→ (u, 1),2 f : S3 → S3 × S3 : u 7→ (1, u),3 f : S3 → S3 × S3 : u 7→ (u, u),4 f : S3 → S3 × S3 : u 7→ (u, ui),5 f : S3 → S3 × S3 : u 7→ (u−1, uiu−1),6 f : S3 → S3 × S3 : u 7→ (uiu−1, u−1).

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Round sphere

(2θ1, 2θ2, 2θ3) = ( 4π3 ,

4π3 ,

4π3 )

(2θ1, 2θ2, 2θ3) = ( 2π3 ,

2π3 ,

2π3 )

(2θ1, 2θ2, 2θ3) = (0, 0, 0)

Berger sphere

(2θ1, 2θ2, 2θ3) = ( 4π3 ,

4π3 ,

4π3 )

(2θ1, 2θ2, 2θ3) = ( 2π3 ,

2π3 ,

2π3 )

(2θ1, 2θ2, 2θ3) = (0, 0, 0)

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Theorem B (B. Dioos, L. Vrancken, X. Wang, 2016)

Let M be a Lagrangian submanifold of constant sectional curvature in the nearly Kähler S3 × S3. Thenup to an isometry of the nearly Kähler S3 × S3, M is locally congruent with one of the followingimmersions:

1 f : S3 → S3 × S3 : u 7→ (u, 1),2 f : S3 → S3 × S3 : u 7→ (1, u),3 f : S3 → S3 × S3 : u 7→ (u, u),4 f : S3 → S3 × S3 : u 7→ (uiu−1, uju−1),5 f : R3 → S3 × S3 : (u, v ,w) 7→ (p(u,w), q(u, v)), where p and q are constant mean curvature

tori in S3.

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Results

Theorem 1

Let f : M −→ S3 × S3 be a Lagrangian immersion into a nearly Kähler manifold S3 × S3 given byf = (p, q) with the angle functions θi . Then, f : M −→ S3 × S3 given by f = (q, p) is also a Lagrangianimmersion with the angle functions θi such that θi = π − θi .

Theorem 2

Let f : M −→ S3 × S3 be a Lagrangian immersion defined by f = (p, q) with the angle functions θi .Then, f? : M −→ S3 × S3 defined by f? = (p, qp) is also a Lagrangian immersion with the anglefunctions θ?i such that θ?i = 2π

3 − θi .

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Lemma 3

Let M be a Lagrangian submanifold of the nearly Kähler manifold S3 × S3 with constant anglefunctions θi .

i. If M is a non-totally geodesic submanifold, then the nonzero components of ωkij are given by

ω312 =

√3

6−

cos (θ2 − θ3)

sin (θ2 − θ3)h3

12, (14)

ω123 =

√3

6+

cos (θ1 − θ3)

sin (θ1 − θ3)h3

12, (15)

ω231 =

√3

6−

cos (θ1 − θ2)

sin (θ1 − θ2)h3

12. (16)

ii. The Codazzi equations of the submanifold M are as followings:

Ei(h312) = 0, i = 1, 2, 3, (17)

h312

(2(ω2

13 + ω321) +

1√

3

)=

13

sin(2(θ1 − θ2)), (18)

h312

(2(ω3

12 + ω231) −

1√

3

)=

13

sin(2(θ1 − θ3)), (19)

h312

(2(ω3

21 + ω132) +

1√

3

)=

13

sin(2(θ2 − θ3)). (20)

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iii. The Gauss equations of the submanifold M are given by

512

+13

cos(2(θ1 − θ2)) − (h312)2 = −ω3

21ω213 + ω3

12ω231 − ω

321ω

231, (21)

512

+13

cos(2(θ1 − θ3)) − (h312)2 = −ω2

31ω312 + ω2

13ω321 − ω

231ω

321, (22)

512

+13

cos(2(θ2 − θ3)) − (h312)2 = −ω1

32ω321 + ω1

23ω312 − ω

132ω

312. (23)

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Theorem 4

A Lagrangian submanifold of the nearly Kähler manifold S3 × S3 with constant angle functions is eithertotally geodesic or has constant sectional curvature in S3 × S3.

Case 1 h312 = 0, a totally geodesic Lagrangian submanifold of the nearly Kähler S3 × S3

Case 2 h312 is a nonzero constant.

i. h312 = − 1

2 , (2θ1, 2θ2, 2θ3) = (0, 2π3 ,

4π3 ),

a flat Lagrangian submanifold.ii. h3

12 = − 14 , (2θ1, 2θ2, 2θ3) = ( 4π

3 ,2π3 , 0),

Lagrangian submanifold with sectional curvature 316 .

iii. h312 = 1

4 , (2θ1, 2θ2, 2θ3) = (0, 2π3 ,

4π3 ),

Lagrangian submanifold with sectional curvature 316 .

iv. h312 = 1

2 , (2θ1, 2θ2, 2θ3) = ( 4π3 ,

2π3 , 0),

a flat Lagrangian submanifold.

Corollary 5

A Lagrangian submanifold of the nearly Kähler manifold S3 × S3 with constant angle functions is locallycongruent to the immersions given in Theorem A and Theorem B.

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References

B.-Y.Chen and K. Oguie, On totally real submanifolds, Trans. Amer. Math. Soc. 193(1974),257–266.

A. Gray, Nearly Kähler manifolds, J. Differential Geometry, 4(1970), 283–309.

P.-A. Nagy, Nearly Kähler geometry and Riemannian folitaions, Asian J. Math., 6(2002),481–504.

J.-B. Butruille, Homogeneous nearly Kähler manifolds, in: Handbook of Pseudo–RiemannianGeometry and Supersymmetry, 399–423, RMA Lect. Math. Theor. Phys., 16, Eur. Math. Soc.Zürich, 2010.

T. Friedrich, and R. Grunewald, On the first eigenvalue of the Dirac operator on 6-dimensionalmanifolds, Ann. Global Anal. Geom. 3 (1985), 265–273.

A. Moroianu and U. Semmelmann, Generalized Killing spinors and Lagrangian graphs, Differ.Geom. Appl. 37(2014), 141–151.

L. Schäfer and K. Smoczyk, Decomposition and minimality of Lagrangian submanifolds innearly Kähler manifolds, Ann. Global Anal. Geom. 37 (2009), no. 3, 221–240.

Y. Zhang, Z. Hu, B. Dioos, L. Vrancken and X. Wang, Lagrangian submanifolds in the6–dimensional nearly Kähler manifolds with parallel second fundamental form, J. Geom. Phys.submitted.

B. Dioos, L. Vrancken and X. Wang, Lagrangian submanifolds in the nearly Kähler S3 × S3,2016, arXiv:1604.05060.

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