Conformal Rigidity

  • Manfredo do Carmo
  • Marcos Dajczer
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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 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Manfredo do Carmo
    • 1
  • Marcos Dajczer
    • 1
  1. 1.Instituto de Matemática Pura e AplicadaRio de JaneiroBrasil

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