Almost-contact manifold

Short description: Geometric structure on a smooth 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 $J_p\circ J_p=-{\mathrm{id}}_{Q_p}$), 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 $(\xi ,\eta ,\varphi )$, where $\xi$ is a vector field on $M$, $\eta$ is a 1-form on $M$, and $\varphi$ is a (1,1)-tensor field on $M$, such that they satisfy the two conditions$${\varphi }^2=-\mathrm{i}\mathrm{d}+\eta \otimes \xi ,\eta (\xi )=1.$$Or in more detail, for any $p\in M$ and any $v\in T_{p}M$,

• ${\eta }_p(v){\xi }_p={\varphi }_p\circ {\varphi }_p(v)+v$
• ${\eta }_p({\xi }_p)=1$

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 ${\varphi }_p:T_{p}M\to T_{p}M$ by$$\begin{array}{rl}{\eta }_p(u)& =0\text{ if }u\in Q_p\\ {\eta }_p({\xi }_p)& =1\\ {\varphi }_p(u)& =J_p(u)\text{ if }u\in Q_p\\ {\varphi }_p(\xi )& =0.\end{array}$$and one can check directly, by decomposing $v$ relative to the direct sum decomposition $T_{p}M=Q_p\oplus \{k{\xi }_p:k\in \mathbb{ℝ}\}$, that$$\begin{array}{rl}{\eta }_p(v){\xi }_p& ={\varphi }_p\circ {\varphi }_p(v)+v\end{array}$$for any $v$ in $T_{p}M$.

In another direction, given $(\xi ,\eta ,\varphi )$, one can define $Q_p$ to be the kernel of the linear map ${\eta }_p$, and one can check that the restriction of ${\varphi }_p$ 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:

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

Relation to other manifolds

Metric

Given an almost-contact manifold equipped with the previously defined $(Q,J,\xi ,\eta ,\varphi )$, 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:$$g(\varphi X,\varphi Y)=g(X,Y)-\eta (X)\eta (Y)\text{ for all }X,Y\in \mathrm{\Gamma }(TM).$$Such a manifold is called an almost contact metric manifold.

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

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

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

Classification

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

Let $(\varphi ,\xi ,\eta ,g)$ be an almost contact metric structure on a $(2n+1)$-manifold, and let $\mathrm{\Phi }(X,Y)=g(X,\varphi Y)$. At each point, regard$$T:=\mathrm{\nabla }\mathrm{\Phi }\in C(V)\subset {\otimes }^3{T}^{*}M,$$where $$\begin{array}{rl}C(V)=\{a\mid a(X,Y,Z)& =-a(X,Z,Y),\\ a(X,\varphi Y,\varphi Z)& =-a(X,Y,Z)+\eta (Y)a(X,\xi ,Z)+\eta (Z)a(X,Y,\xi )\}\end{array}$$For $n>2$, it splits into orthogonal, irreducible, $U(n)\times 1$-invariant subspaces$$C(V)=C_1\oplus C_2\oplus \cdots \oplus C_{12}.$$An almost contact metric manifold is of class $U\subset {⨁}_{i=1}^{12}{C}_i$ if $T_x\in U$ for all $x$. Hence there are $2^{12}$ classes.

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

Specific cases named in the literature:

• Cosymplectic: $U=\{0\}(d\eta =0,d\mathrm{\Phi }=0,\mathrm{\nabla }\mathrm{\Phi }=0)$.
• Nearly $K$-cosymplectic: $U=C_1$.
• Almost cosymplectic: $U=C_2\oplus C_9$.
• $\alpha$-Kenmotsu: $U=C_5$.
• $\alpha$-Sasakian: $U=C_6$.
• Trans-Sasakian $(\alpha ,\beta ):U=C_5\oplus C_6$.
• Quasi-Sasakian: $U=C_6\oplus C_7$.

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 $\eta \wedge {\omega }^n\ne 0$ 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 $\theta :=\sum _i{p}_{i}d{q}_i,\omega :=d\theta ,\eta :=dt$ provides is an almost-contact structure on the manifold $\mathbb{ℝ}\times M$.

The construction of the almost-contact metric structure:

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

To show it, note that$$\eta (\xi )=1,\eta \circ \varphi =0,{{\varphi }^2|}_Q=J^2=-{\mathrm{i}\mathrm{d}}_Q,{\varphi }^2(\xi )=0.$$Thus ${\varphi }^2=-\mathrm{i}\mathrm{d}+\eta \otimes \xi$ on all of $TM$. Hence $(\xi ,\eta ,\varphi )$ is an almost-contact structure.

References

• Blair, David E. (2010) (in en). Riemannian Geometry of Contact and Symplectic Manifolds. Progress in Mathematics. 203 (2nd ed.). Boston, MA: Birkhäuser. pp. 343. doi:10.1007/978-0-8176-4959-3. ISBN 978-0-8176-4958-6. https://link.springer.com/book/10.1007/978-0-8176-4959-3. 
• 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.