Infinitesimal variations of submanifolds

TL;DR

This paper develops a fundamental theorem for infinitesimal variations of Euclidean submanifolds of arbitrary dimension and codimension, establishing integrability conditions and exploring rigidity results for Riemannian product submanifolds.

Contribution

It introduces a fundamental theorem for infinitesimal variations of submanifolds, including integrability conditions and rigidity results, extending classical theories to more general settings.

Findings

Established a fundamental theorem for infinitesimal variations.

Derived integrability conditions involving tensor equations.

Proved rigidity results for Riemannian product submanifolds.

Abstract

This paper deals with the subject of infinitesimal variations of Euclidean submanifolds with arbitrary dimension and codimension. The main goal is to establish a Fundamental theorem for these geometric objects. Similar to the theory of isometric immersions in Euclidean space, we prove that a system of three equations for a certain pair of tensors are the integrability conditions for the differential equation that determines the infinitesimal variations. In addition, we give some rigidity results when the submanifold is intrinsically a Riemannian product of manifolds.