The Obata sphere theorems on a quaternionic contact manifold of dimension bigger than seven

TL;DR

This paper establishes quaternionic contact analogs of Obata's sphere theorems, characterizing manifolds with minimal eigenvalues of the sub-Laplacian as being qc equivalent to the 3-Sasakian sphere.

Contribution

It extends Obata's sphere theorems to quaternionic contact manifolds of dimension greater than seven, including compact and non-compact cases, and introduces a qc Liouville theorem.

Findings

Minimal eigenvalue characterizes the 3-Sasakian sphere up to qc homothety.

Complete non-compact qc manifolds with traceless horizontal Hessian are qc equivalent to the sphere.

A qc Liouville theorem for conformal maps on the sphere is established.

Abstract

We prove a quaternionic contact versions of the Obata's sphere theorems. We show that if the first positive eigenvalue of the sub-Laplacian on a compact qc manifold of dimension bigger than seven takes the smallest possible value then, up to a homothety of the qc structure, the manifold is qc equivalent to the standard 3-Sasakian sphere. We also give a version of the theorem on non-compact qc manifold which is complete with respect to the associated Riemannian metric using the existence of a function with traceless horizontal Hessian. A qc version of the Liouville theorem is shown for qc-conformal maps between open connected sets of the 3-Sasakian sphere.