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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].

##### MSC:
 53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related $$n$$-spaces 53A05 Surfaces in Euclidean and related spaces
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