Contact pairs (bicontact manifolds)

Transcription

1 Contact pairs (bicontact manifolds) Gianluca Bande Università degli Studi di Cagliari XVII Geometrical Seminar, Zlatibor 6 September 2012 G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

2 Basic notions Definition A contact pair (CP) of type (h, k) on an even dimensional manifold M is a pair (α 1,α 2 ) of 1-forms such that: G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

3 Basic notions Definition A contact pair (CP) of type (h, k) on an even dimensional manifold M is a pair (α 1,α 2 ) of 1-forms such that: α 1 (dα 1 ) h α 2 (dα 2 ) k is a volume form (dim M = 2h+2k+ 2) G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

4 Basic notions Definition A contact pair (CP) of type (h, k) on an even dimensional manifold M is a pair (α 1,α 2 ) of 1-forms such that: α 1 (dα 1 ) h α 2 (dα 2 ) k is a volume form (dim M = 2h+2k+ 2) (dα 1 ) h+1 = 0 (α 1 is of constant class 2h+1) G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

5 Basic notions Definition A contact pair (CP) of type (h, k) on an even dimensional manifold M is a pair (α 1,α 2 ) of 1-forms such that: α 1 (dα 1 ) h α 2 (dα 2 ) k is a volume form (dim M = 2h+2k+ 2) (dα 1 ) h+1 = 0 (α 1 is of constant class 2h+1) (dα 2 ) k+1 = 0 (α 2 is of constant class 2k+ 1) G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

6 Characteristic foliations and Reeb vector fields On a manifold M, endowed with a CP, there exist two vector fields (Reeb vector fields) Z 1 and Z 2, uniquely determined by the conditions α 1 (Z 1 ) = α 2 (Z 2 ) = 1, α 1 (Z 2 ) = α 2 (Z 1 ) = 0, i Z1 dα 1 = i Z1 dα 2 = i Z2 dα 1 = i Z2 dα 2 = 0. G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

7 Characteristic foliations and Reeb vector fields On a manifold M, endowed with a CP, there exist two vector fields (Reeb vector fields) Z 1 and Z 2, uniquely determined by the conditions α 1 (Z 1 ) = α 2 (Z 2 ) = 1, α 1 (Z 2 ) = α 2 (Z 1 ) = 0, i Z1 dα 1 = i Z1 dα 2 = i Z2 dα 1 = i Z2 dα 2 = 0. The tangent bundle of M splits as TM = TF 1 TF 2 = TG 1 TG 2 V, where we have TF i = ker dα i kerα i, TG i = ker dα i kerα 1 kerα 2, V = RZ 1 RZ 2. G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

8 Characteristic foliations and Reeb vector fields On a manifold M, endowed with a CP, there exist two vector fields (Reeb vector fields) Z 1 and Z 2, uniquely determined by the conditions α 1 (Z 1 ) = α 2 (Z 2 ) = 1, α 1 (Z 2 ) = α 2 (Z 1 ) = 0, i Z1 dα 1 = i Z1 dα 2 = i Z2 dα 1 = i Z2 dα 2 = 0. The tangent bundle of M splits as TM = TF 1 TF 2 = TG 1 TG 2 V, where we have TF i = ker dα i kerα i, TG i = ker dα i kerα 1 kerα 2, V = RZ 1 RZ 2. Observe that TF 1 e TF 2 are integrable and they determine the so-called characteristic foliations. The subbundles TG i are symplectic and non-integrable. G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

9 Some examples (R 2h+2k+2,α 1,α 2 ) where α 1, α 2 are the Darboux forms on R 2h+1 and R 2k+1 respectively. G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

10 Some examples (R 2h+2k+2,α 1,α 2 ) where α 1, α 2 are the Darboux forms on R 2h+1 and R 2k+1 respectively. On a 6-dimensional (nilpotent) Lie group whose structure equations are the following dω 3 = dω 6 = 0, dω 2 = ω 5 ω 6, dω 1 = ω 3 ω 4, dω 4 = ω 3 ω 5, dω 5 = ω 3 ω 6, G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

11 Some examples (R 2h+2k+2,α 1,α 2 ) where α 1, α 2 are the Darboux forms on R 2h+1 and R 2k+1 respectively. On a 6-dimensional (nilpotent) Lie group whose structure equations are the following dω 3 = dω 6 = 0, dω 2 = ω 5 ω 6, dω 1 = ω 3 ω 4, dω 4 = ω 3 ω 5, dω 5 = ω 3 ω 6, the pair (ω 1,ω 2 ) gives rise to a left invariant CP of type (1, 1). G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

12 Some examples (R 2h+2k+2,α 1,α 2 ) where α 1, α 2 are the Darboux forms on R 2h+1 and R 2k+1 respectively. On a 6-dimensional (nilpotent) Lie group whose structure equations are the following dω 3 = dω 6 = 0, dω 2 = ω 5 ω 6, dω 1 = ω 3 ω 4, dω 4 = ω 3 ω 5, dω 5 = ω 3 ω 6, the pair (ω 1,ω 2 ) gives rise to a left invariant CP of type (1, 1). The product of two contact manifolds and in particular the product of two unit cotangent bundles give examples of CP. G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

13 Motivations and links A CP of type(h, 0) determines a locally conformally symplectic structure. G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

14 Motivations and links A CP of type(h, 0) determines a locally conformally symplectic structure. The existence of a (non-kähler) Vaisman structure on a manifolds is equivalent to the existence of a normal metric CP of type (h, 0). G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

15 Motivations and links A CP of type(h, 0) determines a locally conformally symplectic structure. The existence of a (non-kähler) Vaisman structure on a manifolds is equivalent to the existence of a normal metric CP of type (h, 0). Taking symplectic pairs, in some cases one can do a Boothy-Wang construction to obtain a CP. Recently the symplectic pairs have been used in the study of Almost Kähler manifolds. G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

16 f-structures Definition A CP-structure on M is a triple (α 1,α 2,φ), where (α 1,α 2 ) is a CP and φ is an endomorphism field such that: φ 2 = Id +α 1 Z 1 +α 2 Z 2, φz 1 = φz 2 = 0 where Z 1 and Z 2 are the Reeb vector fields of (α 1,α 2 ). G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

17 f-structures Definition A CP-structure on M is a triple (α 1,α 2,φ), where (α 1,α 2 ) is a CP and φ is an endomorphism field such that: φ 2 = Id +α 1 Z 1 +α 2 Z 2, φz 1 = φz 2 = 0 where Z 1 and Z 2 are the Reeb vector fields of (α 1,α 2 ). Remark For every CP there always exists such a φ and this determines an f -structure in the sense Yano. G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

18 f-structures Definition A CP-structure on M is a triple (α 1,α 2,φ), where (α 1,α 2 ) is a CP and φ is an endomorphism field such that: φ 2 = Id +α 1 Z 1 +α 2 Z 2, φz 1 = φz 2 = 0 where Z 1 and Z 2 are the Reeb vector fields of (α 1,α 2 ). Remark For every CP there always exists such a φ and this determines an f -structure in the sense Yano. Definition The endomorphismφ is said to be decomposable if φ(tf i ) TF i, for i = 1, 2 G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

19 Riemannian metrics On a manifolds M, endowed with a CP-structure (α 1,α 2,φ), it is natural to consider the following kind of metrics Definition A Riemannian metric g is said to be associated if for i = 1, 2 and for all X, Y Γ(TM). g(x,φy) = (dα 1 + dα 2 )(X, Y) g(x, Z i ) = α i (X) G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

20 Riemannian metrics On a manifolds M, endowed with a CP-structure (α 1,α 2,φ), it is natural to consider the following kind of metrics Definition A Riemannian metric g is said to be associated if for i = 1, 2 and for all X, Y Γ(TM). g(x,φy) = (dα 1 + dα 2 )(X, Y) g(x, Z i ) = α i (X) Definition (α 1,α 2,φ, g) will be called MCP (metric contact pair). G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

21 Almost complex structures and normal CP To a CP-structure (α 1,α 2,φ) one can associate two almost complex structure as follows G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

22 Almost complex structures and normal CP To a CP-structure (α 1,α 2,φ) one can associate two almost complex structure as follows J ± = φ±( α 2 Z 1 +α 1 Z 2 ) G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

23 Almost complex structures and normal CP To a CP-structure (α 1,α 2,φ) one can associate two almost complex structure as follows J ± = φ±( α 2 Z 1 +α 1 Z 2 ) Definition A CP-structure (α 1,α 2,φ) will be called normal if J + and J are both integrable. G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

24 Almost complex structures and normal CP To a CP-structure (α 1,α 2,φ) one can associate two almost complex structure as follows J ± = φ±( α 2 Z 1 +α 1 Z 2 ) Definition A CP-structure (α 1,α 2,φ) will be called normal if J + and J are both integrable. J + and J being both integrable is equivalent to the following condition [φ,φ](x, Y)+2dα 1 (X, Y)Z 1 + 2dα 2 (X, Y)Z 2 = 0. G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

25 Remark The leaves of the characteristic foliations of an MCP are minimal submanifolds. G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

26 Remark The leaves of the characteristic foliations of an MCP are minimal submanifolds. For an MCP (α 1,α 2,φ, g), the endomorphismφ is decomposable if and only if the characteristic foliations F 1,F 2 are orthogonal. In this case (α i,φ, g) restricts to a metric contact structure (α i,φ i, g) on the leaves of F j (j i). G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

27 Remark The leaves of the characteristic foliations of an MCP are minimal submanifolds. For an MCP (α 1,α 2,φ, g), the endomorphismφ is decomposable if and only if the characteristic foliations F 1,F 2 are orthogonal. In this case (α i,φ, g) restricts to a metric contact structure (α i,φ i, g) on the leaves of F j (j i). Moreover, if the MCP is normal, then the leaves are Sasakian. G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

28 Vaisman manifolds The existence a contact pair (α,β) of type (h, 0), is equivalent to the existence a Locally Conformally Symplectic structure ω as follows: ω = dα+α β G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

29 Vaisman manifolds The existence a contact pair (α,β) of type (h, 0), is equivalent to the existence a Locally Conformally Symplectic structure ω as follows: ω = dα+α β When the C.P. is also normal metric, then the manifold is Vaisman. G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

30 Vaisman manifolds The existence a contact pair (α,β) of type (h, 0), is equivalent to the existence a Locally Conformally Symplectic structure ω as follows: ω = dα+α β When the C.P. is also normal metric, then the manifold is Vaisman. In dimension 4 there is a complete classification due to F. Belgun. G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

31 An example with not totally geodesic leaves Let us consider the following nilpotent Lie algebra: dω 3 = dω 6 = 0, dω 2 = ω 5 ω 6, dω 1 = ω 3 ω 4, dω 4 = ω 3 ω 5, dω 5 = ω 3 ω 6, G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

32 An example with not totally geodesic leaves Let us consider the following nilpotent Lie algebra: dω 3 = dω 6 = 0, dω 2 = ω 5 ω 6, dω 1 = ω 3 ω 4, dω 4 = ω 3 ω 5, dω 5 = ω 3 ω 6, with CP of type (1, 1) given by (ω 1,ω 2 ) and associated metric g = ω 2 1 +ω ωi 2. i=3 G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

33 An example with not totally geodesic leaves Let us consider the following nilpotent Lie algebra: dω 3 = dω 6 = 0, dω 2 = ω 5 ω 6, dω 1 = ω 3 ω 4, dω 4 = ω 3 ω 5, dω 5 = ω 3 ω 6, with CP of type (1, 1) given by (ω 1,ω 2 ) and associated metric g = ω 2 1 +ω ωi 2. The leaves of the characteristic foliations are minimal but not totally geodesics. i=3 G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

34 Some classification results - Curvature vanishing on the vertical bundle Theorem (B. - Blair - Hadjar) Let M be a (2h+2k+2)-dimensional manifold endowed with an MCP (α 1,α 2,φ, g) of type (h, k) (h 1) and decomposableφ. If the curvature of g vanishes on the vertical subbundle, that is verifies then M is locally isometric to R XY Z 1 = R XY Z 2 = 0, E h+1 S h (4) E k+1 S k (4) if k 1, otherwise to if k = 0. E h+1 S h (4) E 1 G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

35 Some classification results - Locally symmetric normal MCP Theorem (B. - Blair) Let M a manifold endowed with complete, normal, locally symmetric MCP (with decomposable φ). Then the Riemannian universal covering of M is (globally): isometric to a Calabi-Eckmann manifold, S 2m+1 (1) S 2n+1 (1), necessarily compact, or G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

36 Some classification results - Locally symmetric normal MCP Theorem (B. - Blair) Let M a manifold endowed with complete, normal, locally symmetric MCP (with decomposable φ). Then the Riemannian universal covering of M is (globally): isometric to a Calabi-Eckmann manifold, S 2m+1 (1) S 2n+1 (1), necessarily compact, or S 2m+1 (1) R. G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

37 References G. Bande, A. Hadjar Contact pairs Tohoku Math. Journal (2), 57 (2005), no. 2, G. Bande, A. Hadjar On normal contact pairs Internat. J. Math., 21 (2010), no. 6, G. Bande, D. Kotschick Contact pairs and locally conformally symplectic structures Harmonic maps and differential geometry, 85Ð98, Contemp. Math., 542, Amer. Math. Soc., Providence, RI, D.E. Blair, G. D. Ludden and K. Yano Geometry of complex manifolds similar to the Calabi-Eckmann manifolds J. Differential Geom. 9 (1974), G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15

38 References G. Bande, D.E. Blair Symmetry in the Geometry of Metric Contact Pairs preprint. G. Bande, D.E. Blair, A. Hadjar On the Curvature of Metric Contact Pairs to appear in Mediterranean J. of Mathematics. G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds) 6 Settembre / 15