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Intermediate flexibility of surfaces

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  • Mathematics

Abstract

Intermediate flexibility of surfaces COMPOSITIO MATHEMATICA H. G. HELFENSTEIN E. KATZ Intermediate flexibility of surfaces Compositio Mathematica, tome 25, no 1 (1972), p. 71-78. <http://www.numdam.org/item?id=CM_1972__25_1_71_0> © Foundation Compositio Mathematica, 1972, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http:// http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pé- nale. Toute copie ou impression de ce fichier doit contenir la pré- sente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 71 INTERMEDIATE FLEXIBILITY OF SURFACES by H. G. Helfenstein and E. Katz COMPOSITIO MATHEMATICA, Vol. 25, Fasc. 1, 1972, pag. 71-78 Wolters-Noordhoff Publishing Printed in the Netherlands 1. Introduction In Efimov’s article [1 ] cohomology properties of a surface s immersed in Euclidean 3-space are related to existence and classification of in- finitesimal isometric deformations of S. He defines the intermediate flexibility of S with respect to a subgroup F of the 1-dimensional homol- ogy group H of S; it becomes an isometric embedding invariant based on topological properties of S by means of the Rham cohomology. So far, however, this relation between the classical rigidity problems and algebraic topology has remained of a hypothetical nature, since no surface having intermediate flexibility with respect to a non-trivial sub- group was known. We exhibit here the first examples of truly intermediate flexibility. In addition we give a necessary and sufficient condition for surfaces in a certain class to admit intermediate flexibility. Beside the obvious generalization, new phenomena of intermediate flexibility appear in higher dimensions; they will be discussed elsewhere.

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