In this paper the following three goals are addressed. The first goal is to study some strong partial differential equations (PDEs) that imply curvature-flatness, in cases of both symmetric and non-symmetric connection. Although curvature-flatness idea classic for connection, our main theorems about flatness solutions completely new, leaving a while point view geometry entering PDEs. second int...

The study of surfaces immersed in a 3-dimensional ambient space plays a central role in the theory of submanifolds. In addition, Riemannian manifolds with constant sectional curvature can be considered as the most simple examples. Thus, one can think of surfaces with constant Gauss curvature in the Euclidean space R, hyperbolic space H or 3-sphere S as very natural objects of study. Through the...

We prove that any strongly regular Weingarten surface in Euclidean space carries locally geometric principal parameters. The basic theorem states that any strongly regular Weingarten surface is determined up to a motion by its structural functions and the normal curvature function satisfying a geometric differential equation. We apply these results to the special Weingarten surfaces: minimal su...

We extend our previous definition of quasi-local mass to 2-spheres whose Gauss curvature is negative and prove its positivity.

We prove a Gauss-Bonnet formula for the extrinsic curvature of complete surfaces in hyperbolic space under some assumptions on the asymptotic behaviour.

It is shown that the kernels for mean value representations of points in R in terms of the integrals over piecewise smooth hypersurfaces are divergence free vector fields defined by homogeneous functions of degree −(n+1), whose restrictions to the unit sphere are positive and orthogonal to the first harmonics. By Minkowski problem, such a function is the reciprocal of the composition of the Gau...

We consider n-dimensional convex Euclidean hypersurfaces moving with normal velocity proportional to a positive power α of the Gauss curvature. We prove that hypersurfaces contract to points in finite time, and for α ∈ (1/(n + 2], 1/n] we also prove that in the limit the solutions evolve purely by homothetic contraction to the final point. We prove existence and uniqueness of solutions for non-...

We propose a physical model for the capsids of tailed archaeal viruses as viscoelastic membranes under tension. The fluidity is generated by thermal motion of scarlike structures that are an intrinsic feature of the ground state of large particle arrays covering surfaces with nonzero Gauss curvature. The tension is generated by a combination of the osmotic pressure of the enclosed genome and an...

In this paper, by the studying of the Gauss map, Laplacian operator, curvatures of surfaces in R 1 and Bour’s theorem, we are going to identify surfaces of revolution with pointwise 1-type Gauss map property in 3−dimensional Minkowski space. Introduction The classification of submanifolds in Euclidean and Non-Euclidean spaces is one of the interesting topics in differential geometry and in this...