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 R_α on M such that

α(R_α) = 1, ι_Rα dα = 0,

where ι_Rα dα denotes the contraction of dα along R_α. By definition R_α 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 = −I_ξ.

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 g_J [x](·, ·) := κ[x](·, J_x ·) is a J_x–Hermitian inner product on ξ_x for any x ∈ M.

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

J(R_α) = 0.

Note that such a J satisfies

J² =−I + α ⊗ R_α

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

g := g_J + α ⊗ α —– (2)

Furthermore the Nijenhuis tensor of J is defined by

N_J (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 ∈ C_α(M) satisfies N_J = −dα ⊗ R_α

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

ι_Rα γ = 0, L_Rα γ = 0,

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

dΛ^r_B(M) ⊂ Λ^r+1_B(M)

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

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

J_φ(X₁,…, X_r) = φ(JX₁,…, JX_r)

Consequently Λ^r_B(M) ⊗ C splits as

Λ^r_B(M) ⊗ C = ⊕_p+q=r Λ^p,q_J(ξ)

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

∇^ξ_X Y = (∇_X Y)^ξ if X ∈ ξ

= [R_α, Y] if X = R_α

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

∇^ξ_X J = 0, ∇^ξ_Xg_J = 0, ∇^ξ_X dα = 0, ∇^ξ_X Y − ∇^ξ_Y X = [X,Y]^ξ,

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

The basic cohomology class

c^B₁(M) = 1/2π [ρ^T] ∈ H^1,1_B(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) dim_RL = 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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