A symmetry result on submanifolds of space forms and Applications

TL;DR

This paper establishes a symmetry theorem for codimension-one submanifolds in space forms, utilizing geodesic distance and normal curvature, with applications to sphere characterization in Kähler manifolds with toric actions.

Contribution

It introduces a new symmetry result for submanifolds in space forms and applies it to characterize spheres in Kähler manifolds with toric symmetry.

Findings

Proved a symmetry theorem relating geodesic distance and normal curvature.

Derived sphere characterization theorems for Kähler manifolds with toric actions.

Enhanced understanding of submanifold geometry in space forms.

Abstract

In this paper we prove a symmetry result on submanifolds of codimension one in a n + 1-dimensional space form, related to the geodesic distance function and to the normal curvature of some fixed vector field. As applications we will prove sphere characterization type theorems for Kahler manifolds endowed with a toric group action.