‎let $m^n$ be an $n(ngeq 3)$-dimensional complete connected and‎ ‎oriented spacelike hypersurface in a de sitter space or an anti-de‎ ‎sitter space‎, ‎$s$ and $k$ be the squared norm of the second‎ ‎fundamental form and gauss-kronecker curvature of $m^n$‎. ‎if $s$ or‎ ‎$k$ is constant‎, ‎nonzero and $m^n$ has two distinct principal‎ ‎curvatures one of which is simple‎, ‎we obtain some‎ ‎charact...

We discuss notions of Gauss curvature and mean curvature for polyhedral surfaces. The discretizations are guided by the principle of preserving integral relations for curvatures, like the Gauss–Bonnet theorem and the mean-curvature force balance equation.

It is fair to say that Riemannian geometry started with Gauss’s famous ”Disquisitiones generales” from 1827 in which one finds a rigorous discussion of what we now call the Gauss curvature of a surface. Much has been written about the importance and influence of this paper, see in particular the article [Do] by P.Dombrowski for a careful discussion of its contents and influence during that time...

We investigate the existence of Taub-NUT/bolt solutions in Gauss-Bonnet gravity and obtain the general form of these solutions in d dimensions. We find that for all nonextremal NUT solutions of Einstein gravity having no curvature singularity at r = N , there exist NUT solutions in Gauss-Bonnet gravity that contain these solutions in the limit that the Gauss-Bonnet parameter α goes to zero. Fur...

Conventional physical dogma, justified by the local success of Newtonian dynamics for particles, assigns a Euclidean metric with signature (plus, plus, plus) to the three spatial dimensions. Minimal surfaces are of zero mean curvature and negative Gauss curvature in a Euclidean space, which supports affine evolutionary processes. However, experimental evidence now indicates that the non-affine ...

We study the mean curvature flow of complete space-like submanifolds in pseudo-Euclidean space with bounded Gauss image, as well as that of complete submanifolds in Euclidean space with convex Gauss image. By using the confinable property of the Gauss image under the mean curvature flow we prove the long time existence results in both cases. We also study the asymptotic behavior of these soluti...

We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski space into the hyperbolic space. As applications, we prove a Bernstein theorem which says that if the image of the Gauss map is bounded from one side, then the spacelike constant mean curvature hypersurface must be linear. This result extends the previous theorems obtained...

In this paper, we survey recent results on Gauss-Bonnet-Chern formulae and related issues for closed Riemannian manifolds with variable curvature. Among other things, we address the following problem: “if M is an oriented 2n-dimensional closed manifold with non-positive curvature, then is it true that its Euler number χ(M) satisfies the inequality (−1)χ(M) ≥ 0?” We will present some partial ans...

LetX be a compact, strictly convex C-hypersurface in the (n+1)-dimensional Euclidean space R. The Gauss map ofX maps the hypersurface one-to-one and onto the unit n-sphere S. One may parametrize X by the inverse of the Gauss map. Consequently, the Gauss curvature can be regarded as a function on S. The classical Minkowski problem asks conversely when a positive function K on S is the Gauss curv...