Conformal Rigidity

Manfredo do Carmo and Marcos Dajczer
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Manfredo do Carmo: Instituto de Matemática Pura e Aplicada
Marcos Dajczer: Instituto de Matemática Pura e Aplicada

A chapter in Manfredo P. do Carmo – Selected Papers, 2012, pp 267-289 from Springer

Abstract: Abstract In one of his less known papers, Cartan [3] studies the conformal deformations of hypersurfaces of an Euclidean space $$R^{n+1},n>4.$$ As a consequence of his methods, he obtains a (local) sufficient condition for conformal rigidity ([3], pg. 101; see also Corollary 1.3 below). In this paper we obtain a generalization of Cartan’s rigidity theorem for codimension k ≤ 4. This gives a new proof of Cartan’s result that is independent of the methods of [3]. The fact that we have restricted ourselves to codimensions k ≤ 4 seems to be a technical point, and we will return to that in a while. As a simple consequence of our methods, we obtain an improvement, for codimension k ≤ 5, of Allendoerfer’s isometric rigidity theorem [1].

Keywords: Bilinear Form; Fundamental Form; Normal Bundle; Null Vector; Main Lemma (search for similar items in EconPapers)
Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-25588-5_21

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DOI: 10.1007/978-3-642-25588-5_21

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