We find a kind of variations of Gauss-Codazzi-Ricci equations suitable for Kaluza-Klein reduction and Cauchy problem. Especially the counterpart of extrinsic curvature tensor has antisymmetric part as well as symmetric one. If the dependence of metric tensor on reduced dimensions is negligible it becomes a pure antisymmetric tensor. PACS:03.70;11.15

We prove a discrete Gauss-Bonnet-Chern theorem ∑ g∈V K(g) = χ(G) for finite graphs G = (V,E), where V is the vertex set and E is the edge set of the graph. The dimension of the graph, the local curvature form K and the Euler characteristic are all defined graph theoretically.

We derive estimates of the Hessian of two smooth functions defined on Grassmannian manifold. Based on it, we can derive curvature estimates for minimal submanifolds in Euclidean space via Gauss map as [24]. In this way, the result for Bernstein type theorem done by Jost and the first author could be improved.

The helicoidal surface is a generalization of rotation surface in a Minkowski space. We study helicoidal surfaces in a Minkowski 3-space in terms of their Gauss map and provide some examples of new classes of helicoidal surfaces with constant mean curvature in a Minkowski 3-space.

It is proven results about existence and nonexistence of unit normal sections submanifolds the Euclidean space sphere, which associated Gauss maps, are harmonic. Some applications to constant mean curvature hypersurfaces sphere isoparametric obtained too.

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 ...

Abstract. By a classical result of Kazdan-Warner, for any smooth signchanging function f with negative mean on the torus (M,gb) there exists a conformal metric g = egb Gauss curvature Kg = f , which can be obtained from a minimizer u of Dirichlet’s integral in a suitably chosen class of functions. As shown by Galimberti, these minimizers exhibit “bubbling” in a certain limit regime. Here we sha...

We consider natural conformal invariants arising from the Gauss-Bonnet formulas on manifolds with boundary, and study conformal deformation problems associated to them. The purpose of this paper is to study conformal deformation problems associated to conformal invariants on manifolds with boundary. From analysis point of view, the problem becomes a non-Dirichlet boundary value problems for ful...

A class of vacuum initial-data sets is described which are based on certain expressions for the extrinsic curvature first studied and employed by Bowen and York. These expressions play a role for the momentum constraint of General Relativity which is analogous to the role played by the Coulomb solution for the Gauss-law constraint of electromagnetism.

We construct, for any " good " Cantor set F of S n−1 , an immersion of the sphere S n with set of points of zero Gauss-Kronecker curvature equal to F ×D 1 , where D 1 is the 1-dimensional disk. In particular these examples show that the theorem of Matheus-Oliveira strictly extends two results by do Carmo-Elbert and Barbosa-Fukuoka-Mercuri.