Prescribed Gauss-Kronecker curvature problems are widely studied in the literature. Famous among them is the Minkowski problem. It was studied by H. Minkowski, A.D. Alexandrov, H. Lewy, A.V. Pogorelov, L. Nirenberg and at last solved by S.Y. Cheng and S.T. Yau [CY]. After that, V.I.Oliker [O] researched the arbitrary hypersurface with prescribed Gauss curvature in Euclidean space. On the other ...

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.

We introduce the generalized helical hypersurface having a space-like axis in five-dimensional Minkowski space. compute first and second fundamental form matrices, Gauss map, shape operator matrix of hypersurface. Additionally, we curvatures by using Cayley–Hamilton theorem. Moreover, give some relations for mean Gauss–Kronecker Finally, obtain Laplace–Beltrami

A production function f is called quasi-sum if there are continuous strict monotone functions F, h1, . . . , hn with F > 0 such that f(x) = F (h1(x1) + · · · + hn(xn)) (cf. [1]). A quasi-sum production function is called quasi-linear if at most one of F, h1, . . . , hn is a nonlinear function. For a production function f , the graph of f is called the production hypersurface of f . In this pape...

In this paper we extend the well known results on the existence and regularity of solutions of the Dirichlet problem for Monge-Ampère equations in a strictly convex domain to an arbitrary smooth bounded domain in Rn as well as in a general Riemannian manifold. We prove for the nondegenerate case that a sufficient (and necessary) condition for the classical solvability is the existence of a subs...

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...