# Umbilical spacelike submanifolds of arbitrary co-dimension

## TL;DR

This paper characterizes umbilical spacelike submanifolds of any co-dimension in semi-Riemannian manifolds using the total shear tensor, defining shear and umbilical spaces and establishing their dimension relation.

## Contribution

It provides necessary and sufficient conditions for umbilical submanifolds via the total shear tensor and introduces the concepts of shear and umbilical spaces with a dimension relation.

## Key findings

- Dimension of shear space plus umbilical space equals co-dimension.
- Characterization of umbilical submanifolds using the total shear tensor.
- Framework applicable to arbitrary co-dimension submanifolds.

## Abstract

Given a semi-Riemannian manifold, we give necessary and sufficient conditions for a Riemannian submanifold of arbitrary co-dimension to be umbilical along normal directions. We do that by using the so-called \emph{total shear tensor}, i.e., the trace-free part of the second fundamental form. We define the \emph{shear space} and the \emph{umbilical space} as the spaces generated by the total shear tensor and by the umbilical vector fields, respectively. We show that the sum of their dimensions must equal the co-dimension.