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Bending invariants for hypersurfaces. (English) Zbl 0978.53013
Séminaire de théorie spectrale et géométrie. Année 1998-1999. St. Martin D’Hères: Université de Grenoble I, Institut Fourier, Sémin. Théor. Spectrale Géom., Chambéry-Grenoble. 17, 105-109 (1999).
From the introduction: A very long-standing problem in classical differential geometry is the bendability problem for compact hypersurfaces in Euclidean spaces. By a hypersurface we mean a connected Riemannian \(n\)-manifold \((n\geq 2)\) which is \(C^2\)-isometrically immersed in \(\mathbb{R}^{n+1}\). A hypersurface \(M\) is said to be bendable if it can be isometrically and non-trivially deformed in \(\mathbb{R}^{n+1}\). It is said rigid if every hypersurface which is \(C^2\) isometric to \(M\) is congruent to \(M\) (i.e. is obtained by applying an ambient rigid motion). The bendability problem is to decide whether there exist bendable compact hypersurfaces or not. In this paper, we are concerned with a closely related problem. We are interested in geometric quantities defined on the hypersurface which are extrinsic – that is, depend on the way the surface is immersed in \(\mathbb{R}^{n+1}\) and not only on the metric on \(M\) – but are invariant under isometric deformation of \(M\). Results in this direction may be viewed as unbendability results in a weak sense.
For the entire collection see [Zbl 0929.00011].

53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53A05 Surfaces in Euclidean and related spaces
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