Infinitesimal rigidity of Euclidean submanifolds

Dajczer, Marcos; Rodríguez, Lucio

doi:10.5802/aif.1242

Cited by 16 publications

(15 citation statements)

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“…Alexandrov's annuli r is globally rigid. For the case of n ≥ 3, Dajczer-Rodriguez [5] proved Theorem 8. If the rank of the matrix (h ij ) is greater than 2, where h = h ij dx i dx j is the second fundamental form, then the hypersurface is globally and infinitesimally rigid.…”

Section: Gauss-codazzi Equations and Darboux Equationmentioning

confidence: 97%

The Rigidity of Hypersurfaces in Euclidean Space

2019

Chin. Ann. Math. Ser. B

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Section: Gauss-codazzi Equations and Darboux Equationmentioning

confidence: 97%

The Rigidity of Hypersurfaces in Euclidean Space

2019

Chin. Ann. Math. Ser. B

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“…Dajczer and Rodríguez [19] showed that submanifolds in low codimension are generically infinitesimally rigid, that is, generically only trivial infinitesimal variations are possible. In fact, they proved that certain algebraic conditions on the second fundamental form of an immersion, known to give isometric rigidity, yield infinitesimal rigidity as well.…”

Section: Prefacementioning

confidence: 99%

Infinitesimal variations of submanifolds

Dajczer

Jimenez

2021

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The main purpose of these lecture notes is to present recent results in the theory of infinitesimal variations of submanifolds. The smooth variations under consideration are infinitesimally isometric or, in greater generality, infinitesimally conformal. The concept of infinitesimal variation is the infinitesimal analogue of an isometric variation and refers to smooth variations that preserve lengths "up to the first order". In the more general case of conformal infinitesimal variations, lengths are preserved similarly but now up to a conformal factor. The study of such variations is realized by means of the corresponding variational vector fields, called infinitesimal bendings and conformal infinitesimal bendings respectively. Hence, these lecture notes contain results about nontrivial infinitesimal bendings and the geometry of the submanifolds that carry them.

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“…where the last steps follow using (5) and the assumption on β. Thus the tensors L i are well defined on M i , 1 ≤ i ≤ r. Moreover, since B is symmetric, these tensors verify…”

Section: Bending Of a Product Of Manifoldsmentioning

confidence: 99%

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 Euclidean submanifolds

Cited by 16 publications

References 1 publication

The Rigidity of Hypersurfaces in Euclidean Space

The Rigidity of Hypersurfaces in Euclidean Space

Infinitesimal variations of submanifolds

Infinitesimal Bendings

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