Vol. 317, No. 1, 2022

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Genuine infinitesimal bendings of submanifolds

Marcos Dajczer and Miguel Ibieta Jimenez

Vol. 317 (2022), No. 1, 119–141
Abstract

A basic question in submanifold theory is whether a given isometric immersion f : Mn n+p of a Riemannian manifold of dimension n 3 into Euclidean space with low codimension admits, locally or globally, a genuine infinitesimal bending. That is, if there exists a genuine smooth variation of f by immersions that are isometric up to the first order. Until now only the hypersurface case p = 1 is well understood. We show that a strong necessary local condition to admit such a bending is the submanifold to be ruled and give a lower bound for the dimension of the rulings. In the global case, we describe the situation of compact submanifolds of dimension n 5 in codimension p = 2.

Keywords
Euclidean submanifolds, infinitesimal bendings
Mathematical Subject Classification
Primary: 53A07, 53B25, 53C40
Milestones
Received: 18 December 2019
Accepted: 8 January 2022
Published: 19 June 2022
Authors
Marcos Dajczer
IMPA
Rio de Janeiro
Brazil
Miguel Ibieta Jimenez
Universidade de São Paulo
São Carlos
Brazil