Geometrical versus Topological Properties of Manifolds and a Remark on Poincaré Conjecture
Krerley Oliveira | Matheus, Carlos
Immersions | Hausdorff dimension | finite geometrical type
Given a compact $n$-dimensional immersed Riemannian manifold $M^n$ 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 $S^n$. A consequence of our main theorems is a conjecture which is equivalent to Poincaré Conjecture. Also, we define a concept of finite geometrical type and prove that finite geometrical type hypersurfaces are topologically the sphere minus a finite number of points. A characterization of the $2n$-catenoid is obtained.