# Normal curvature of pseudo-umblical submanifolds in a sphere

## TL;DR

This paper investigates conditions under which a compact pseudo-umbilical submanifold in a sphere must be totally geodesic, linking normal curvature, scalar curvature, and second fundamental form.

## Contribution

It establishes new criteria involving normal curvature and scalar curvature that guarantee a pseudo-umbilical submanifold is totally geodesic.

## Key findings

- Normal curvature, scalar curvature, and second fundamental form conditions imply total geodesicity
- Provides criteria for identifying totally geodesic submanifolds in spheres
- Enhances understanding of geometric properties of pseudo-umbilical submanifolds

## Abstract

Let M be a compact pseudo-umbilical submanifold of the unit sphere S. In the present note, it is shown that if the normal curvature, scalar curvature S and square of the length of second fundamental form satisfy certain conditions, then M is totally geodesic.