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.