Differential equations on contact Riemannian manifolds

Building on a work by K. Yano, [28], we study certain differential equations on a compact Riemannian manifold whose almost CR structure is not integrable, in general. We prove a "universality" property of Tanno's equation $\nabla_\xi \mathcal{L}_\xi g - 2 (\mathcal{L}_\xi g) \cdot \phi = 0$. We show that the sublaplacian $\Delta_H$ (introduced by S. Tanno, cf. op. cit.) is subelliptic of order $1/2$ (hence $\Delta_H$ is hypoelliptic and has a discrete spectrum tending to $+ \infty$. We consider the tangential Cauchy-Riemann equation $\overline{\partial}_H \omega = 0$, $\omega \in \Omega^{0,q} (M)$, $q \geq 0$, and associate a twisted cohomology (cf. I. Vaisman, [31]) with the corresponding tangential Cauchy-Riemann pseudocomplex. We build a Lorentzian metric $G_\eta$ on the total space of a certain principal $S^1$-bundle $\pi : F(M) \to M$ over a contact manifold $(M, \eta)$. When the almost CR structure of $M$ is integrable, $G_\eta$ is the Fefferman metric (cf. J.M. Lee, [2]) of $(M, - \eta )$. We show that a $C^\infty$ map $f: M \to N$ of a contact Riemannian manifold $M$ into a Riemannian manifold $(N, g)$ satisfies $\Delta_H f^i + 4 g^{\alpha \bar \beta} \xi_\alpha (f^j) \xi_{\bar \beta} (f^k) (\Gamma^i_{jk} \circ f) = 0$ if and only if the vertical lift $f \circ \pi$ of $f$ is a harmonic map with respect to the (generalized) Fefferman matric $G_\eta$.