The Lichnerowicz–Obata Theorem on Sub-Riemannian Manifolds with Transverse Symmetries

Abstract

We prove a lower bound for the first eigenvalue of the sub-Laplacian on sub-Riemannian manifolds with transverse symmetries. When the manifold is of $$H$$ -type, we obtain a corresponding rigidity result: If the optimal lower bound for the first eigenvalue is reached, then the manifold is equivalent to a 1- or a 3-Sasakian sphere.