Contact Geometry and Manifolds


Let M be a manifold of dimension 2n + 1. A contact structure on M is a distribution ξ ⊂ TM of dimension 2n, such that the defining 1-form α satisfies

α ∧ (dα)n ≠ 0 —– (1)

A 1-form α satisfying (1) is said to be a contact form on M. Let α be a contact form on M; then there exists a unique vector field Rα on M such that

α(Rα) = 1, ιRα dα = 0,

where ιRα dα denotes the contraction of dα along Rα. By definition Rα is called the Reeb vector field of the contact form α. A contact manifold is a pair (M, ξ) where M is a 2n + 1-dimensional manifold and ξ is a contact structure. Let (M, ξ) be a contact manifold and fix a defining (contact) form α. Then the 2-form κ = 1/2 dα defines a symplectic form on the contact structure ξ; therefore the pair (ξ, κ) is a symplectic vector bundle over M. A complex structure on ξ is the datum of J ∈ End(ξ) such that J2 = −Iξ.

Let α be a contact form on M, with ξ = ker α and let κ = 1/2 dα. A complex structure J on ξ is said to be κ-calibrated if gJ [x](·, ·) := κ[x](·, Jx ·) is a JxHermitian inner product on ξx for any x ∈ M.

The set of κ-calibrated complex structures on ξ will be denoted by Cα(M). If J is a complex structure on ξ = ker α, then we extend it to an endomorphism of TM by setting

J(Rα) = 0.

Note that such a J satisfies

J2 =−I + α ⊗ Rα

If J is κ-calibrated, then it induces a Riemannian metric g on M given by

g := gJ + α ⊗ α —– (2)

Furthermore the Nijenhuis tensor of J is defined by

NJ (X, Y) = [JX, JY] − J[X, JY] − J[Y, JX] + J2[X, Y] for any X, Y ∈ TM

A Sasakian structure on a 2n + 1-dimensional manifold M is a pair (α, J), where

• α is a contact form;

• J ∈ Cα(M) satisfies NJ = −dα ⊗ Rα

The triple (M, α, J) is said to be a Sasakian manifold. Let (M, ξ) be a contact manifold. A differential r-form γ on M is said to be basic if

ιRα γ = 0, LRα γ = 0,

where L denotes the Lie derivative and Rα is the Reeb vector field of an arbitrary contact form defining ξ. We will denote by ΛrB(M) the set of basic r-forms on (M, ξ). Note that

rB(M) ⊂ Λr+1B(M)

The cohomology HB(M) of this complex is called the basic cohomology of (M, ξ). If (M, α, J) is a Sasakian manifold, then

J(ΛrB(M)) = ΛrB(M), where, as usual, the action of J on r-forms is defined by

Jφ(X1,…, Xr) = φ(JX1,…, JXr)

Consequently ΛrB(M) ⊗ C splits as

ΛrB(M) ⊗ C = ⊕p+q=r Λp,qJ(ξ)

and, according with this gradation, it is possible to define the cohomology groups Hp,qB(M). The r-forms belonging to Λp,qJ(ξ) are said to be of type (p, q) with respect to J. Note that κ = 1/2 dα ∈ Λ1,1J(ξ) and it determines a non-vanishing cohomology class in H1,1B(M). The Sasakian structure (α, J) also induces a natural connection ∇ξ on ξ given by

ξX Y = (∇X Y)ξ if X ∈ ξ

= [Rα, Y] if X = Rα

where the subscript ξ denotes the projection onto ξ. One easily gets

ξX J = 0, ∇ξXgJ = 0, ∇ξX dα = 0, ∇ξX Y − ∇ξY X = [X,Y]ξ,

for any X, Y ∈ TM. Consequently we have Hol(∇ξ) ⊆ U(n).

The basic cohomology class

cB1(M) = 1/2π [ρT] ∈ H1,1B(M)

is called the first basic Chern class of (M, α, J) and, if it vanishes, then (M, α, J) is said to be null-Sasakian.

Furthermore a Sasakian manifold is called α-Einstein if there exist λ, ν ∈ C(M, R) such that

Ric = λg + να ⊗ α, where Ric is the Ricci Tensor.

A submanifold p: L ֒→ M of a 2n + 1-dimensional contact manifold (M, ξ) is said to be Legendrian if :

1) dimRL = n,

2) p(TL) ⊂ ξ

Observe that, if α is a defining form of the contact structure ξ, then condition 2) is equivalent to say that p(α) = 0. Hence Legendrian submanifolds are the analogue of Lagrangian submanifolds in contact geometry.

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