Pseudohermitian Structures and Contact Metric Structures

Question

Let $M$ be an odd-dimensional manifold. An almost contact metric structure on $M$ is a 4-tuple $(\xi, \eta, \phi, g)$, where $\xi$ is a vector field, $\eta$ a one-form, $\phi$ an endomorphism of the tangent bundle, and $g$ a Riemannian metric, all enjoying the following conditions:

\begin{aligned} \phi^2 &= -I +\xi\otimes \eta\\ \eta(\xi) &= 1\\ g(X,Y) &= g(\phi X, \phi Y) + \eta(X)\eta(Y). \end{aligned}

An almost contact metric structure is a contact metric structure if the following integrability condition is satisfied: \begin{equation} g(X,\phi Y)=d\eta(X,Y). \end{equation}

In CR geometry, contact metric structures appear naturally; if $(M,\eta)$ is a pseudohermitian manifold, then it defines a contact metric structure, with $\xi$ the Reeb vector field, $\phi$ the complex structure on the holomorphic tangent bundle, and $g$ the Webster metric.

Questions: Is there an example of a contact metric structure on some odd-dimensional manifold that does not come from a pseudohermitian structure? Is there a class of manifolds for which pseudohermitian structures are in bijection with contact metric structures?

Answer 1

Consider the (1,2) tensor Q defined by

$Q_{jk}^i=\nabla_k \phi_j^i+\xi^i\phi_j^r\nabla_k \eta_r+\phi_r^i \nabla_k \xi^r \eta_j$

Define $J$ by restricting $\phi$ to the kernel of $\eta$. Then $ (M,\eta,J)$ is a strictly pseudoconvex, integrable CR manifold if and only if $Q=0$. This result was proved by Tanno, this is Proposition 2.1 in

Variational problems on contact Riemannian manifolds, Trans. Amer. Math. Soc. 314 (1989), 349-379