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arXiv · 1303.0409

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

Stefan Ivanov·Alexander Petkov·Dimiter Vassilev
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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.

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Stefan Ivanov, Alexander Petkov, Dimiter Vassilev. 2013-03-02. The Obata sphere theorems on a quaternionic contact manifold of dimension bigger than seven. https://arxiv.org/abs/1303.0409

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