Question

I wonder if Morse Theory on pseudo-Hermitian manifold is developed. For example, I wonder if the following statement on pseudo-Hermitian manifold, which is corresponding to the Riemannian case, is true:

Suppose $f$ is a smooth real-valued non-degenerate function on a pseudo-Hermitian manifold $(M,\theta)$, $a < b$, $f^{−1}[a, b]$ is compact, and there are no critical values between $a$ and $b$ in the sense that for all $x\in f^{−1}[a, b]$, the subgradient of $f$ at $x$ is not zero. Then $M^a$ is diffeomorphic to $M^b$, and $M^b$ deformation retracts onto $M^a$, where $M^a=f^{-1}(-\infty,a]$.

Answer 1

Checking the various definition of "pseudo-Hermitian manifolds" in on-line papers, it seems they are in particular Finsler manifolds (there is a norm on the $T_x M$, continuously depending on $x$ ). In this case, one may apply the usual Lusternik-Schnirel'man theory (see e.g. Lusternik-Schnirelman theory on Banach manifolds, Topology 5 (1966), 115–132, by Richard Palais). In particular, the quoted statement follows.