In the mathematical field of differential geometry, an almost-contact structure is a certain kind of geometric structure on a smooth manifold, obtained by combining a contact-element structure (not necessarily a contact structure) and an almost-complex structure. They can be considered as an odd-dimensional counterpart to almost complex manifolds.

They were introduced by John Gray in 1959. Shigeo Sasaki in 1960 introduced Sasakian manifold to study them.

Given a smooth manifold ${\displaystyle M}$, an almost-contact structure is a triple ${\displaystyle (Q,J,\xi )}$ of a hyperplane distribution ${\displaystyle Q}$, an almost-complex structure ${\displaystyle J}$ on ${\displaystyle Q}$, and a vector field ${\displaystyle \xi }$ which is transverse to ${\displaystyle Q}$. That is, for each point ${\displaystyle p}$ of ${\displaystyle M}$, one selects a contact element (that is, a codimension-one linear subspace ${\displaystyle Q_{p}}$ of the tangent space ${\displaystyle T_{p}M}$), a linear complex structure on it (that is, a linear function ${\displaystyle J_{p}:Q_{p}\to Q_{p}}$ such that ${\displaystyle J_{p}\circ J_{p}=-\operatorname {id} _{Q_{p}}}$), and an element ${\displaystyle \xi _{p}}$ of ${\displaystyle T_{p}M}$ which is not contained in ${\displaystyle Q_{p}}$. As usual, the selection must be smooth.

Equivalently, one may define an almost-contact structure as a triple ${\displaystyle (\xi ,\eta ,\phi )}$, where ${\displaystyle \xi }$ is a vector field on ${\displaystyle M}$, ${\displaystyle \eta }$ is a 1-form on ${\displaystyle M}$, and ${\displaystyle \phi }$ is a (1,1)-tensor field on ${\displaystyle M}$, such that they satisfy the two conditions$${\displaystyle \phi ^{2}=-\mathrm {id} +\eta \otimes \xi ,\quad \eta (\xi )=1.}$$Or in more detail, for any ${\displaystyle p\in M}$ and any ${\displaystyle v\in T_{p}M}$,

Because the choice of the transverse vector field ${\displaystyle \xi }$ is smooth, the field ${\displaystyle \xi }$ is a co-orientation of the distribution of contact elements ${\displaystyle Q}$.

More abstractly, it can be defined as a G-structure obtained by reduction of the structure group from ${\displaystyle GL(2n+1)}$ to ${\displaystyle U(n)\times 1}$.

In one direction, given ${\displaystyle (\xi ,J,Q)}$, one can define for each ${\displaystyle p}$ in ${\displaystyle M}$ a linear map ${\displaystyle \eta _{p}:T_{p}M\to \mathbb {R} }$ and a linear map ${\displaystyle \phi _{p}:T_{p}M\to T_{p}M}$ by$${\displaystyle {\begin{aligned}\eta _{p}(u)&=0{\text{ if }}u\in Q_{p}\\\eta _{p}(\xi _{p})&=1\\\phi _{p}(u)&=J_{p}(u){\text{ if }}u\in Q_{p}\\\phi _{p}(\xi )&=0.\end{aligned}}}$$and one can check directly, by decomposing ${\displaystyle v}$ relative to the direct sum decomposition ${\displaystyle T_{p}M=Q_{p}\oplus \left\{k\xi _{p}:k\in \mathbb {R} \right\}}$, that$${\displaystyle {\begin{aligned}\eta _{p}(v)\xi _{p}&=\phi _{p}\circ \phi _{p}(v)+v\end{aligned}}}$$for any ${\displaystyle v}$ in ${\displaystyle T_{p}M}$.

In another direction, given ${\displaystyle (\xi ,\eta ,\phi )}$, one can define ${\displaystyle Q_{p}}$ to be the kernel of the linear map ${\displaystyle \eta _{p}}$, and one can check that the restriction of ${\displaystyle \phi _{p}}$ to ${\displaystyle Q_{p}}$ is valued in ${\displaystyle Q_{p}}$, thereby defining ${\displaystyle J_{p}}$.