Rigidity Theorems for Complete Sasakian Manifolds with Constant Pseudo-Hermitian Scalar Curvature

TL;DR

This paper establishes rigidity theorems for complete Sasakian manifolds with constant pseudo-Hermitian scalar curvature, characterizing special geometric structures using subelliptic inequalities and maximum principles.

Contribution

It derives new subelliptic differential inequalities and proves rigidity results that characterize Sasakian pseudo-Einstein manifolds and Sasakian space forms under certain curvature conditions.

Findings

Rigidity theorems for Sasakian pseudo-Einstein manifolds.

Characterization of Sasakian space forms among pseudo-Einstein manifolds.

Use of subelliptic estimates and maximum principles in proofs.

Abstract

The orthogonal decomposition of the Webster curvature provides us a way to characterize some canonical metrics on a pseudo-Hermitian manifold. We derive some subelliptic differential inequalities from the Weitzenb\"ock formulas for the traceless pseudo-Hermitian Ricci tensor and the Chern-Moser tensor of Sasakian manifolds with constant pseudo-Hermitian scalar curvature and Sasakian pseudo-Einstein manifolds respectively. By means of either subelliptic estimates or maximum principle, some rigidity theorems are established to characterize Sasakian pseudo-Einstein manifolds among Sasakian manifolds with constant pseudo-Hermitian scalar curvature and Sasakian space forms among Sasakian pseudo-Einstein manifolds respectively.