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.

中文翻译：

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子流形的无穷小变化

本文涉及具有任意维数和余维数的欧几里得子流形的无穷小变化的主题。主要目标是为这些几何对象建立基本定理。与欧几里德空间中的等距浸入理论类似，我们证明了对于确定无穷小变化的微分方程的可积条件是针对某一对张量的三个方程组。此外，当子流形本质上是流形的黎曼乘积时，我们给出了一些刚性结果。