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 J^{2} = −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

J^{2} =−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] + J^{2}[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

*of an arbitrary contact form defining ξ. We will denote by Λ*

**Reeb vector field**^{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}(X_{1},…, X_{r}) = φ(JX_{1},…, 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}_{X}g_{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^{B}_{1}(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_{R}L = 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.