Geometrical versus Topological Properties of Manifolds

Details Geometrical versus Topological Properties of Manifolds Geometrical versus Topological Properties of Manifolds

2003-07-04

arxiv.org/abs/math/0307070v2

Journal of the Institute of Mathematics of Jussieu, vol. 4, n. 4, p. 630-651, 2005

Mathematics

Differential Geometry

Scientific paper

Given a compact $n$-dimensional immersed Riemannian manifold $M^n$ in some Euclidean space we prove that if the Hausdorff dimension of the singular set of the Gauss map is small, then $M^n$ is homeomorphic to the sphere $S^n$. Also, we define a concept of finite geometrical type and prove that finite geometrical type hypersurfaces with small set of points of zero Gauss-Kronecker curvature are topologically the sphere minus a finite number of points. A characterization of the $2n$-catenoid is obtained.

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