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Conformal Rigidity

In: Manfredo P. do Carmo – Selected Papers

Author

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  • Manfredo do Carmo

    (Instituto de Matemática Pura e Aplicada)

  • Marcos Dajczer

    (Instituto de Matemática Pura e Aplicada)

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].

Suggested Citation

  • Manfredo do Carmo & Marcos Dajczer, 2012. "Conformal Rigidity," Springer Books, in: Keti Tenenblat (ed.), Manfredo P. do Carmo – Selected Papers, edition 127, pages 267-289, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-25588-5_21
    DOI: 10.1007/978-3-642-25588-5_21
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