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 $M^a=f^{-1}(-\infty,a]$.