C-R Immersions and Sub-Riemannian Geometry

On any strictly pseudoconvex CR manifold $M$, of CR dimension $n$, equipped with a positively oriented contact form $\theta$, we consider natural $\epsilon$-contractions i.e. contractions $g_\epsilon^M$ of the Levi form $G_\theta$, such that the norm of the Reeb vector field $T$ of $(M, \theta )$ is of order $O(\epsilon^{-1})$. We study isopseudohermitian (i.e. $f^\ast \Theta = \theta$) Cauchy-Riemann immersions $f : M \to (A, \Theta )$ between strictly pseudoconvex CR manifolds $M$ and $A$, where $\Theta$ is a contact form on $A$. For every contraction $g_\epsilon^A$ of the Levi form $G_\Theta$ we write the embedding equations for the immersion $f : M \to ( A, \, g_\epsilon^A )$. A pseudohermitan version of the Gauss equation for an isopseudohermitian C-R immersion is obtained by an elementary asymptotic analysis as $\epsilon \to 0^+$. For every isopseudohermitian immersion $f : M \to S^{2N+1}$ into a sphere $S^{2N+1} \subset {\mathbb C}^{N+1}$, we show that Webster's pseudhermitian scalar curvature $R$ of $(M, \theta )$ satisfies the inequality $R \leq 2 n [ (f^\ast g_\Theta )(T,T) + n + 1] + (1/29 { || H(f) ||^2_{g_\Theta^f} + || trace_{G_\theta} \Pi_{H(M)} ( \nabla^\top - \nabla ) ||^2_{f^\ast g_\Theta}}$ with equality if and only if $B(f) = 0$ and $\nabla^\top = \nabla$ on $H(M) \otimes H(M)$. This gives a pseudohermitian analog to a classical result by S-S. Chern on minimal isometric immersions into space forms.