# On Pseudo-Umbilical Spacelike Submanifolds in Indefinite Space Form Mn+p p (c)

## TL;DR

This paper establishes that pseudo-umbilical spacelike submanifolds in indefinite space forms are necessarily totally geodesic and totally umbilical, providing new intrinsic inequalities and extending previous results.

## Contribution

It derives an intrinsic inequality for pseudo-umbilical spacelike submanifolds and proves they are both totally geodesic and totally umbilical, advancing understanding of their geometric properties.

## Key findings

- Pseudo-umbilical spacelike submanifolds are totally geodesic.
- Such submanifolds are also totally umbilical.
- The paper introduces an intrinsic inequality for these submanifolds.

## Abstract

In the present note, first we derive an intrinsic inequality for Pseudo-umbilical spacelike submanifold in an indefinite space form. We use this inequality to show that such submanifold is totally geodesic. In the rest part of this paper, using a result of Aiyama [1], we prove that Pseudo-umbilical spacelike subamnifold is totally umbilical.