Comparison theorems in centro-affine differential geometry

Abstract

For weighted manifolds with Bakry-Émery Ricci curvature bounded from below, comparison geometric results have been established. The Bakry-Émery Ricci curvature depends on the effective dimension $N$. In this paper, the case $N = 1$ is further studied in the setting of centro-affine differential geometry. Several examples, such as rotationally symmetric ellipsoids and hyperboloids, arise in rigidity phenomena of comparison theorems.