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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.

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Acknowledgements

This work is the result of the visit 21171/IV/19 funded by the Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia in connection with the “Jiménez De La Espada” Regional Programme For Mobility, Collaboration And Knowledge Exchange. Marcos Dajczer was partially supported by the Fundación Séneca Grant 21171/IV/19 (Programa Jiménez de la Espada), MICINN/FEDER project PGC2018-097046-B-I00, and Fundación Séneca project 19901/GERM/15, Spain. Miguel I. Jimenez thanks the mathematics department of the Universidad de Murcia for the hospitality during his visit where part of this work was developed.

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Correspondence to Marcos Dajczer.

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Dajczer, M., Jimenez, M.I. Infinitesimal Variations of Submanifolds. Bull Braz Math Soc, New Series 52, 573–589 (2021). https://doi.org/10.1007/s00574-020-00220-x

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  • DOI: https://doi.org/10.1007/s00574-020-00220-x

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