The Bochner-Type Formula and The First Eigenvalue of the sub-Laplacian on a Contact Riemannian Manifold

TL;DR

This paper extends the Bochner-type formula and eigenvalue bounds from CR geometry to contact Riemannian manifolds, broadening the understanding of sub-Laplacian spectra in geometric analysis.

Contribution

It introduces a contact Riemannian version of the Bochner formula and generalizes the CR Lichnerowicz theorem for eigenvalue estimates.

Findings

Established a contact Riemannian Bochner-type formula.

Generalized the Lichnerowicz eigenvalue bound to contact Riemannian manifolds.

Provided new tools for spectral analysis in contact geometry.

Abstract

Contact Riemannian manifolds, with not necessarily integrable complex structures, are the generalization of pseudohermitian manifolds in CR geometry. The Tanaka-Webster-Tanno connection on such a manifold plays the role of Tanaka-Webster connection in the pseudohermitian case. We prove the contact Riemannian version of the pseudohermitian Bochner-type formula, and generalize the CR Lichnerowicz theorem about the sharp lower bound for the first nonzero eigenvalue of the sub-Laplacian to the contact Riemannnian case.