A characterization of the extrinsic spheres in a Riemannian manifold

TL;DR

This paper proves a theorem characterizing n-dimensional submanifolds in a Riemannian manifold that locally contain two tangent extrinsic spheres, showing they are either totally geodesic or extrinsic spheres.

Contribution

It provides a new characterization of extrinsic spheres in Riemannian manifolds based on local tangent extrinsic spheres.

Findings

Submanifolds with two tangent extrinsic spheres are either totally geodesic or extrinsic spheres.

The theorem applies to n-dimensional submanifolds with n>2.

It advances understanding of the geometric structure of submanifolds in Riemannian geometry.

Abstract

The following Theorem is proved: Let M be an n-dimensional (n>2) submanifold of a Riemannian manifold N. Suppose that through each point p of M there exist two (n-1)-dimensional extrinsic spheres of N, which are contained in M in a neighbourhood of p and are tangent to each other at p. Then M is totally geodesic in N or an extrinsic sphere of N.