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Another counter example, and somehow more exotic, is the Klein bottle.

Last week I considered again principal curvature (pc) and principal curvature directions (pcd) of a, for the sake of simplicity, 2-manifold embedded in 3-space. In this simple case, the pc and pcd of …

Consider a connected, complete and compact Riemannian manifold $M$. Is it correct that the following equality holds: $\text{inj}(x)=\text{dist}\left(x,\text{CuL}(x)\right)$? Or in words that the injec …

In the book Riemannian geometry - modern introduction by Isaac Chavel, three different definitions of convexity are introduced. I am looking for an example of a set which is convex but not strongly co …

In the paper Surface sampling and the intrinsic Voronoi diagram (2008), Ramsay Dyer defines the Gaussian curvature radius at a point $x$ of a surface $S$ to be $\rho_K(x) = 1/\sqrt{K(x)}$ where $K(x)= …

In his book "Riemannian Geometry" do Carmo cites the Hopf-Rinow theorem in chapter 7. (theorem 2.8). One of the equivalences there deals with the cover of the manifold using nested sequence of compact …

Recently I began to consider algebraic surfaces, that is, the zero set of a polynomial in 3 (or more variables). My algebraic geometry background is poor, and I'm more used to differential and Riemann …