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.^[1] Shigeo Sasaki in 1960 introduced Sasakian manifold to study them.^[2]

Definition

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.^[3]

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 ${\displaystyle v\in T_{p}M}$,

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

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 ${\displaystyle U(n)\times 1}$.

Equivalence

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 ${\displaystyle J_{p}}$.

Properties

Given an almost contact structure on a ${\displaystyle (2n+1)}$-manifold, we have:^{[3]: Theorem 4.1}

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

Relation to other manifolds

Metric

Given an almost-contact manifold equipped with the previously defined ${\displaystyle (Q,J,\xi ,\eta ,\phi )}$, we may add a Riemannian metric ${\displaystyle g}$ to it. We say the metric is compatible with the almost-contact structure iff the metric satisfies the metric compatibility condition:$${\displaystyle g(\phi X,\phi Y)=g(X,Y)-\eta (X)\eta (Y)\quad {\text{ for all }}X,Y\in \Gamma (TM).}$$Such a manifold is called an almost contact metric manifold.^[3]

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:$${\displaystyle g(X,Y)={\frac {1}{2}}(k(X,Y)+k(\phi X,\phi Y)+\eta (X)\eta (Y))}$$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.^[4]

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$${\displaystyle T:=\nabla \Phi \in C(V)\subset \otimes ^{3}T^{*}M,}$$where $${\displaystyle {\begin{aligned}C(V)=\{a\mid a(X,Y,Z)&=-a(X,Z,Y),\\a(X,\phi Y,\phi Z)&=-a(X,Y,Z)+\eta (Y)a(X,\xi ,Z)+\eta (Z)a(X,Y,\xi )\}\end{aligned}}}$$For ${\textstyle n>2}$, it splits into orthogonal, irreducible, ${\textstyle U(n)\times 1}$-invariant subspaces$${\displaystyle C(V)=C_{1}\oplus C_{2}\oplus \cdots \oplus C_{12}.}$$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 ${\displaystyle 2n+1}$ induces an almost-contact structure.^[5] Specifically, a cosymplectic structure is a tuple ${\displaystyle (\eta ,\omega )}$ where ${\displaystyle \eta }$ is a closed 1-form, ${\displaystyle \omega }$ is a closed 2-form, and ${\displaystyle \eta \wedge \omega ^{n}\neq 0}$ at every point. One way to produce a cosymplectic structure is by foliating the manifold into symplectic manifolds, and set ${\displaystyle \omega }$ to be the symplectic structure on each manifold, and have ${\displaystyle \ker \eta }$ parallel to the tangent planes through the foliation.^[6]

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

The construction of the almost-contact metric structure:^{[5]: Theorem 3.3}

• ${\textstyle Q=\operatorname {ker} \eta }$ (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 ${\displaystyle Q}$ consistent with ${\displaystyle \omega ^{n}}$. 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 ${\displaystyle \mathbb {R} ^{2n}}$ has sympletic form ${\displaystyle \omega =\sum _{i=1}^{n}dp_{i}\wedge dq_{i}}$, then ${\displaystyle (q,p)\mapsto (-p,q)}$ 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$${\displaystyle \eta (\xi )=1,\quad \eta \circ \phi =0,\left.\quad \phi ^{2}\right|_{Q}=J^{2}=-\mathrm {id} _{Q},\quad \phi ^{2}(\xi )=0.}$$Thus ${\textstyle \phi ^{2}=-\mathrm {id} +\eta \otimes \xi }$ on all of ${\textstyle TM}$. Hence ${\textstyle (\xi ,\eta ,\phi )}$ is an almost-contact structure.