The local geometric properties such as curvatures and normal vectors play important roles in analyzing the local shape of objects. The result of the geometric operations such as mesh simplification and mesh smoothing is dependent on how to compute the curvatures of meshes, because there is no exact definition of the discrete curvature in meshes. In this paper, we indicate the fatal error in com...

We show that there exists a suitable neighborhood of a constant curvature hyperbolic metric such that, for all initial data in this neighborhood, the corresponding solution to a normalized cross curvature flow exists for all time and converges to a hyperbolic metric. We show that the same technique proves an analogous result for Ricci flow. Additionally, we show short time existence and uniquen...

Spaces of positive curvature play a special role in geometry. Although the class of manifolds with positive (sectional) curvature is expected to be relatively small, so far there are only a few known obstructions. Moreover, for closed simply connected manifolds these coincide with the known obstructions to nonnegative curvature which are: (1) the Betti number theorem of Gromov which asserts tha...