Abstract
The main theorems of this chapter are of global nature and show that complete Euclidean submanifolds with low codimension that allow isometric deformations are rather special. A first basic result in this direction is a theorem due to Sacksteder, which asserts that any compact Euclidean hypersurface \(f\colon M^n\to \mathbb {R}^{n+1}\), n ≥ 3, is isometrically rigid, provided that the subset of totally geodesic points of f does not disconnect M n. Even if that subset disconnects the manifold, only discrete isometric deformations are possible. In fact, any such deformation is a reflection with respect to an affine hyperplane. The corresponding versions of that result for hypersurfaces of the sphere and hyperbolic space are also discussed.
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Dajczer, M., Tojeiro, R. (2019). Deformations of Complete Submanifolds. In: Submanifold Theory . Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-9644-5_13
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DOI: https://doi.org/10.1007/978-1-4939-9644-5_13
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