Morse Theory on pseudo-Hermitian manifold - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T23:20:00Z http://mathoverflow.net/feeds/question/95228 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95228/morse-theory-on-pseudo-hermitian-manifold Morse Theory on pseudo-Hermitian manifold Paul 2012-04-26T04:38:40Z 2012-04-26T08:58:57Z <p>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: </p> <p>Suppose $f$ is a smooth real-valued non-degenerate function on a pseudo-Hermitian manifold $(M,\theta)$, $a &lt; b$, $f^{−1}[a, b]$ 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 $M^a=f^{-1}(-\infty,a]$.</p> http://mathoverflow.net/questions/95228/morse-theory-on-pseudo-hermitian-manifold/95232#95232 Answer by Pietro Majer for Morse Theory on pseudo-Hermitian manifold Pietro Majer 2012-04-26T07:40:08Z 2012-04-26T08:58:57Z <p>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. <em>Lusternik-Schnirelman theory on Banach manifolds</em>, Topology 5 (1966), 115–132, by Richard Palais). In particular, the quoted statement follows.</p>