Infinitesimal rigidity of submanifolds

Goldstein, R. A.; Ryan, Patrick J.

doi:10.4310/jdg/1214432675

Cited by 10 publications

(3 citation statements)

References 4 publications

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“…We choose η of unit length for simplicity. Thus, that ( 16) holds means β(X, Y ), η + α(X, Y ), ξ = 0 (17) for any X, Y ∈ X(M).…”

Section: The Second Local Resultsmentioning

confidence: 99%

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Genuine infinitesimal bendings of submanifolds

Dajczer,

Jimenez

2019

Preprint

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A basic question in submanifold theory is whether a given isometric immersion f : M n → R n+p of a Riemannian manifold of dimension n ≥ 3 into Euclidean space with low codimension p 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 was 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.

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“…We choose η of unit length for simplicity. Thus, that ( 16) holds means β(X, Y ), η + α(X, Y ), ξ = 0 (17) for any X, Y ∈ X(M).…”

Section: The Second Local Resultsmentioning

confidence: 99%

“…In this section, we discuss several properties of a tensor associated to an infinitesimal bending called in the classical theory of surfaces the associated rotation field; for instance see [22]. For basic facts on infinitesimal bendings we refer to [9], [10], [11] and [17].…”

Section: The Associated Tensormentioning

confidence: 99%

Genuine infinitesimal bendings of submanifolds

Dajczer,

Jimenez

2019

Preprint

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“…If for each pair of isometric immersions fI:M--,N and '2:MN" there exists a continuous curve rs, 8 e [0,hi in the group G of isometries of N such that r o id and r a 1 2' we say that M is uniquely continuously isometrically immersed into N [8].…”

Section: Introductionmentioning

confidence: 99%

Unique isometric immersion into hyperbolic space

Beltagy

1992

International Journal of Mathematics and Mathematical Sciences

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Infinitesimal Bendings

Dajczer¹,

Tojeiro²

2019

Universitext

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This paper deals with the subject of infinitesimal bendings 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 of an infinitesimal bending. In addition, we give some rigidity results when the submanifold is intrinsically a Riemannian product of manifolds.Mathematics Subject Classification: 53A07, 53B25

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Infinitesimal rigidity of submanifolds

Cited by 10 publications

References 4 publications

Genuine infinitesimal bendings of submanifolds

Genuine infinitesimal bendings of submanifolds

Unique isometric immersion into hyperbolic space

Infinitesimal Bendings

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