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geometrical versus topological properties of manifolds

Published online by Cambridge University Press:  09 May 2005

carlos matheus  and
krerley oliveira
carlos matheus
Affiliation:
instituto nacional de matemática pura e aplicada, estrada dona castorina, 110 jardim botânico 22460-320, rio de janeiro, rj, brazil (matheus@impa.br)
krerley oliveira
Affiliation:
instituto nacional de matemática pura e aplicada, estrada dona castorina, 110 jardim botânico 22460-320, rio de janeiro, rj, brazil (matheus@impa.br)
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Abstract

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 a 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.

Keywords

hausdorff dimensionfinite geometrical typeimmersionsprimary 53-02
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 4 , Issue 4 , October 2005 , pp. 639 - 651
Copyright
2005 cambridge university press

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