Almost-contact manifold

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.

Definition

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

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

• $\eta _{p}(v)\xi _{p}=\phi _{p}\circ \phi _{p}(v)+v$
• $\eta _{p}(\xi _{p})=1$

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

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

Equivalence

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

In another direction, given $(\xi ,\eta ,\phi )$, 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 $J_{p}$.

Properties

Given an almost contact structure on a $(2n+1)$-manifold, we have:

• $\phi \xi =0$
• $\eta \circ \phi =0$
• $\phi$ has rank 2n.

Relation to other manifolds

Metric

Given an almost-contact manifold equipped with the previously defined $(Q,J,\xi ,\eta ,\phi )$, we may add a Riemannian metric $g$ to it. We say the metric is compatible with the almost-contact structure iff the metric satisfies the metric compatibility condition:$$g(\phi X,\phi Y)=g(X,Y)-\eta (X)\eta (Y)\quad {\text{ for all }}X,Y\in \Gamma (TM).$$Such a manifold is called an almost contact metric manifold.

Define the fundamental 2-form ${\textstyle \Phi }$ by ${\textstyle \Phi (X,Y)=g(X,\phi Y)}$. Then ${\textstyle \Phi }$ is skew-symmetric and ${\textstyle \eta (X)=g(X,\xi )}$.

Compatible metrics are easy to find. That is, they are not rigid. To construct one, take any metric ${\textstyle k^{\prime }}$, and let ${\textstyle k(X,Y)=k^{\prime }\left(\phi ^{2}X,\phi ^{2}Y\right)+\eta (X)\eta (Y)}$, then this is a compatible metric:$$g(X,Y)={\frac {1}{2}}(k(X,Y)+k(\phi X,\phi Y)+\eta (X)\eta (Y))$$Special cases used in the literature are:

• Contact metric manifold: additionally ${\textstyle \eta \wedge (d\eta )^{n}\neq 0}$ and ${\textstyle d\eta =2\Phi }$.
• Sasakian manifold: contact metric manifold, with normality condition ${\textstyle N_{\phi }+2d\eta \otimes \xi =0}$.
• Almost coKähler manifold: almost contact metric, with ${\textstyle d\eta =0}$ and ${\textstyle d\Phi =0}$ (normality not assumed).

Classification

They have been fully classified via group representation theory into 4096 classes.

Let ${\textstyle (\phi ,\xi ,\eta ,g)}$ be an almost contact metric structure on a ${\textstyle (2n+1)}$-manifold, and let ${\textstyle \Phi (X,Y)=g(X,\phi Y)}$. At each point, regard$$T:=\nabla \Phi \in C(V)\subset \otimes ^{3}T^{*}M,$$where $${\begin{aligned}C(V)=\{a\mid a(X,Y,Z)&=-a(X,Z,Y),\\a(X,\phi Y,\phi Z)&=-a(X,Y,Z)+\eta (Y)a(X,\xi ,Z)+\eta (Z)a(X,Y,\xi )\}\end{aligned}}$$For ${\textstyle n>2}$, it splits into orthogonal, irreducible, ${\textstyle U(n)\times 1}$-invariant subspaces$$C(V)=C_{1}\oplus C_{2}\oplus \cdots \oplus C_{12}.$$An almost contact metric manifold is of class ${\textstyle U\subset \bigoplus _{i=1}^{12}C_{i}}$ if ${\textstyle T_{x}\in U}$ for all ${\textstyle x}$. Hence there are ${\textstyle 2^{12}}$ classes.

Given such a manifold, it can be classified as follows: compute ${\textstyle \nabla \Phi }$, project it onto the twelve ${\textstyle C_{i}}$ (via the formulas in Table III of the paper), and identify the class by which ${\textstyle C_{i}}$ components are nonzero.

Specific cases named in the literature:

• Cosymplectic: ${\textstyle U=\{0\}(d\eta =0,d\Phi =0,\nabla \Phi =0)}$.
• Nearly ${\textstyle K}$-cosymplectic: ${\textstyle U=C_{1}}$.
• Almost cosymplectic: ${\textstyle U=C_{2}\oplus C_{9}}$.
• ${\textstyle \alpha }$-Kenmotsu: ${\textstyle U=C_{5}}$.
• ${\textstyle \alpha }$-Sasakian: ${\textstyle U=C_{6}}$.
• Trans-Sasakian ${\textstyle (\alpha ,\beta ):U=C_{5}\oplus C_{6}}$.
• Quasi-Sasakian: ${\textstyle U=C_{6}\oplus C_{7}}$.

Examples

A cosymplectic structure on a smooth manifold of dimension $2n+1$ induces an almost-contact structure. Specifically, a cosymplectic structure is a tuple $(\eta ,\omega )$ where $\eta$ is a closed 1-form, $\omega$ is a closed 2-form, and $\eta \wedge \omega ^{n}\neq 0$ at every point. One way to produce a cosymplectic structure is by foliating the manifold into symplectic manifolds, and set $\omega$ to be the symplectic structure on each manifold, and have $\ker \eta$ parallel to the tangent planes through the foliation.

Another common way to construct a cosymplectic structure is through time-dependent Hamiltonian mechanics. Let a phase space be $M$. A trajectory of a system in phase space is a path in $\mathbb {R} \times M$. Let $p,q$ be canonical coordinates on the phase space, which may be allowed to vary over time. Then $\theta :=\sum _{i}p_{i}dq_{i},\;\omega :=d\theta ,\;\eta :=dt$ provides is an almost-contact structure on the manifold $\mathbb {R} \times M$.

The construction of the almost-contact metric structure:

• ${\textstyle Q=\operatorname {ker} \eta }$ (a rank- ${\textstyle 2n}$ distribution).
• The Reeb field ${\textstyle \xi }$ by ${\textstyle \eta (\xi )=1}$ and ${\textstyle \iota _{\xi }\omega =0}$ (uniquely determined because ${\textstyle \eta \wedge \omega ^{n}\neq 0}$ ).
• Since ${\textstyle \left.\omega \right|_{Q}}$ is symplectic, choose an orientation of $Q$ consistent with $\omega ^{n}$. Then pick any almost-complex structure ${\textstyle J}$ on ${\textstyle Q}$ that is ${\textstyle \omega }$-compatible. In detail, it must satisfy ${\textstyle J^{2}=-\mathrm {id} }$ on ${\textstyle Q}$, ${\textstyle \omega (\cdot ,J\cdot )}$ is a positive-definite bilinear form, and ${\textstyle \omega (J\cdot ,J\cdot )=\omega }$.
  • Explicitly, if $\mathbb {R} ^{2n}$ has sympletic form $\omega =\sum _{i=1}^{n}dp_{i}\wedge dq_{i}$, then $(q,p)\mapsto (-p,q)$ is a ${\textstyle \omega }$-compatible complex form on it.
• Set ${\textstyle \left.\phi \right|_{Q}=J}$ and ${\textstyle \phi (\xi )=0}$.

To show it, note that$$\eta (\xi )=1,\quad \eta \circ \phi =0,\left.\quad \phi ^{2}\right|_{Q}=J^{2}=-\mathrm {id} _{Q},\quad \phi ^{2}(\xi )=0.$$Thus ${\textstyle \phi ^{2}=-\mathrm {id} +\eta \otimes \xi }$ on all of ${\textstyle TM}$. Hence ${\textstyle (\xi ,\eta ,\phi )}$ is an almost-contact structure.

References

• Blair, David E. (2010). Riemannian Geometry of Contact and Symplectic Manifolds. Progress in Mathematics. Vol. 203 (2nd ed.). Boston, MA: Birkhäuser. p. 343. doi:10.1007/978-0-8176-4959-3. ISBN 978-0-8176-4958-6.
• Sasaki, Shigeo (1960). "On differentiable manifolds with certain structures which are closely related to almost contact structure, I". Tohoku Mathematical Journal. 12 (3): 459–476. doi:10.2748/tmj/1178244407.