Comparison theorems in pseudo-Hermitian geometry and applications

TL;DR

This paper extends classical comparison theorems from Riemannian geometry to pseudo-Hermitian manifolds, focusing on geodesics with respect to the Tanaka-Webster connection, and establishes several fundamental geometric results.

Contribution

It generalizes key comparison theorems like Hopf-Rinow, Cartan-Hadamard, and Bonnet-Myers to pseudo-Hermitian geometry using the Tanaka-Webster connection.

Findings

Established Hopf-Rinow type theorem in pseudo-Hermitian setting

Proved Cartan-Hadamard type theorem for pseudo-Hermitian manifolds

Derived Bonnet-Myers type results for pseudo-Hermitian manifolds

Abstract

In this paper, we study the theory of geodesics with respect to the Tanaka-Webster connection in a pseudo-Hermitian manifold, aiming to generalize some comparison results in Riemannian geometry to the case of pseudo-Hermitian geometry. Some Hopf-Rinow type, Cartan-Hadamard type and Bonnet-Myers type results are established.