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Intermediate flexibility of surfaces

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  • Law
  • Mathematics


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. <> © Foundation Compositio Mathematica, 1972, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http:// implique l’accord avec les conditions générales d’utilisation ( 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 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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