On deformable hypersurfaces in space forms

Abstract

[Bil ]. A few years after SBRANA, CARTAN (1) published similar results but in the language of envelopes of hyperplanes.Deformable hypersurfaces can be divided into four classes. Submanifolds belonging to the two less interesting ones, namely, surface-like and ruled hypersurfaces, are highly deformable. On the contrary, while hypersurfaces in one of the remaining classes admit, precisely, a continuous one-parameter family of isometric deformations, elements belonging to the other class have a unique one.The main result in the Sbrana-Cartan theory is a parametric classification of all hypersurfaces in the two most important classes. For the first one, the description turns out to be quite satisfactory in the sense that it enables the construction of many explicit examples; cf. p 22 of [Ca1] and w 4 of this paper. For the remaining class the situation is quite different. First of all, the parametric description is cumbersome. One has to search for surfaces for which a pair of Christoffel symbols associated to a conjugate 9 system of coordinates satisfies a certain complicated system of second order partial differential equations. Thus, it is not surprising that no example comes out from this re-(*) Entrata in Redazione il 10 febbraio 1997 e, in versione riveduta, il 4 giugno 1997. Indirizzo degli AA.: M. DAJCZER: IMPA,

The infinitesimally bendable Euclidean hypersurfaces

“…Note that if M n is simply-connected then the set of all isometric immersions of M n into R n+1 consists of ruled immersions with the same rulings; see [5]. Moreover, this set can be parametrized by the set of all smooth functions in an interval.…”

Genuine Isometric and Conformal Deformations of Submanifolds

“…A modern account of the classification of Sbrana-Cartan hypersurfaces was given in, 12 where also some questions left open in the works by Sbrana and Cartan were answered, as the existence of hypersurfaces of the discrete type and the possibility of gluing together hypersurfaces of distinct classes.…”

Euclidean Hypersurfaces with Genuine Conformal Deformations in Codimension Two

“…The classification of Sbrana-Cartan hypersurfaces was extended to the case of nonflat ambient space forms by Dajczer-Florit-Tojeiro [9]. Moreover, among other things, in that paper it was given an affirmative answer to the question of whether Sbrana-Cartan hypersurfaces that allow a single deformation do exist, which was not addressed by Sbrana and Cartan. A nonparametric description of Cartan hypersurfaces of dimension n ≥ 5 was given in [13], where it was shown that any such hypersurface arises by intersecting the light-cone V n+2 in Lorentzian space L n+3 with a flat space-like submanifold of codimension two of L n+3 .…”

“…Moreover, among other things, in that paper it was given an affirmative answer to the question of whether Sbrana-Cartan hypersurfaces that allow a single deformation do exist, which was not addressed by Sbrana and Cartan. A nonparametric description of Cartan hypersurfaces of dimension n ≥ 5 was given in [13], where it was shown that any such hypersurface arises by intersecting the light-cone V n+2 in Lorentzian space L n+3 with a flat space-like submanifold of codimension two of L n+3 . We refer to [9] and [13], respectively, for modern accounts of the classifications of Sbrana-Cartan and Cartan hypersurfaces.…”