Almost-contact manifold

Template:Short description Template:No footnotes In the mathematical field of differential geometry, an almost-contact structure is a certain kind of geometric structure on a smooth manifold. Such structures were introduced by Shigeo Sasaki in 1960.

Precisely, given a smooth manifold $M$, an almost-contact structure consists of a hyperplane distribution $Q$, an almost-complex structure $J$ on $Q$, and a vector field $\xi$ which is transverse to $Q$. That is, for each point $p$ of $M$, one selects a codimension-one linear subspace $Q_p$ of the tangent space $T_{p}M$, a linear map $J_p:Q_p\to Q_p$ such that $J_p\circ J_p=-{\mathrm{id}}_{Q_p}$, and an element ${\xi }_p$ of $T_{p}M$ which is not contained in $Q_p$.

Given such data, one can define, for each $p$ in $M$, a linear map ${\eta }_p:T_{p}M\to ℝ$ and a linear map ${\phi }_p:T_{p}M\to T_{p}M$ by $$\begin{array}{cc}{\eta }_p(u)& =0\text{ if }u\in Q_p\\ {\eta }_p({\xi }_p)& =1\\ {\phi }_p(u)& =J_p(u)\text{ if }u\in Q_p\\ {\phi }_p(\xi )& =0.\end{array}$$ This defines a one-form $\eta$ and (1,1)-tensor field $\phi$ on $M$, and one can check directly, by decomposing $v$ relative to the direct sum decomposition $T_{p}M=Q_p\oplus \{k{\xi }_p:k\in ℝ\}$, that $$\begin{array}{cc}{\eta }_p(v){\xi }_p& ={\phi }_p\circ {\phi }_p(v)+v\end{array}$$ for any $v$ in $T_{p}M$. Conversely, one may define an almost-contact structure as a triple $(\xi ,\eta ,\phi )$ which satisfies the two conditions

• ${\eta }_p(v){\xi }_p={\phi }_p\circ {\phi }_p(v)+v$ for any $v\in T_{p}M$
• ${\eta }_p({\xi }_p)=1$

Then one can define $Q_p$ to be the kernel of the linear map ${\eta }_p$, and one can check that the restriction of ${\phi }_p$ to $Q_p$ is valued in $Q_p$, thereby defining $J_p$.

References

• David E. Blair. Riemannian geometry of contact and symplectic manifolds. Second edition. Progress in Mathematics, 203. Birkhäuser Boston, Ltd., Boston, MA, 2010. xvi+343 pp. Template:ISBN, Template:Doi Template:Closed access
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