Related papers: The rigidity of hypersurface in Euclidean space

We prove a topological rigidity theorem for closed hypersurfaces of the Euclidean sphere and of an elliptic space form. It asserts that, under a lower bound hypothesis on the absolute value of the principal curvatures, the hypersurface is…

We consider closed biharmonic hypersurfaces in the Euclidean sphere and prove a rigidity result under a suitable condition on the scalar curvature. Moreover, we establish an integral formula involving the position vector for biharmonic…

In this paper, we investigate the rigidity problems of complete hypersurfaces with constant mean curvature and constant scalar curvature in Euclidean spaces. Firstly, under some conditions of Gaussian-Kronecker curvature, we provide…

Conditions, related to the so-called bending problem are considered for hypersurfaces of a pseudo-Euclidean space. Corresponding theorems are proved.

Since $n$-dimensional $\lambda$-hypersurfaces in the Euclidean space $\mathbb {R}^{n+1}$ are critical points of the weighted area functional for the weighted volume-preserving variations, in this paper, we study the rigidity properties of…

In this paper, we prove a rigidity theorem for smooth strictly convex domains in Euclidean spaces.

In this paper, we characterize round spheres in the Euclidean space under some suitable conditions on the r-mean curvature.

In this paper, we first give some new characterizations of geodesic spheres in the hyperbolic space by the condition that hypersurface has constant weighted shifted mean curvatures, or constant weighted shifted mean curvature ratio, which…

In this paper, our purpose is to study rigidity theorems for $\lambda$-hypersurfaces in Euclidean space under Gauss map. As a Bernstein type problem for $\lambda$-hypersurfaces, we prove that an entirely graphic $\lambda$-hypersurface in…

In 1972, E. P. Senkin generalized the celebrated theorem of A. V. Pogorelov on unique determination of compact convex surfaces by their intrinsic metrics in the Euclidean 3-space $E^3$ to higher dimensional Euclidean spaces $E^{n+1}$ under…

We study the curvature condition which uniquely characterizes the hemisphere. In particular, we prove the Min-Oo conjecture for hypersurfaces in Euclidean space and hyperbolic space.

In this paper, we study strong r-helix hypersurfaces and the special curves on these surfaces. Moreover, we investigated the relations between strong r-helix hypersurfaces and the Gauss transformations of these surfaces in Euclidean…

We prove rigidity for hypersurfaces with boundary in the unit $(n+1)$-sphere with scalar curvature bounded below by $n(n-1)$. Under appropriate boundary conditions, the hypersurfaces are shown to be part of the equatorial spheres. The lower…

In this paper we present a rigidity theorem for locally isometric hypersurfaces with a curvature restriction in de Sitter space. This is an analogue to the case for Riemannian space forms given by Guan and Shen in [5].

The main aim of this paper is to investigate Darboux rectifying curves on a smooth surface immersed in the Euclidean space. First, we discuss the component of the position vector of a Darboux rectifying curve on a smooth immersed surface…

In this paper, we first investigate several rigidity problems for hypersurfaces in the warped product manifolds with constant linear combinations of higher order mean curvatures as well as "weighted'' mean curvatures, which extend the work…

In this paper, we give some new characterizations of umbilic hypersurfaces in general warped product manifolds, which can be viewed as generalizations of the work in \cite{KLP18} and \cite{WX14}. Firstly, we prove the rigidity for…

The purpose of this paper is, first, to give an algorithm that enables to obtain the lines of curvature on parametric hypersurfaces in Euclidean 4-space, and then, to obtain the curvatures of such lines by using the extended Darboux frame…

A hypersurface without umbilics in the n+1 dimensional Euclidean space is known to be determined by the Moebius metric and the Moebius second fundamental form up to a Moebius transformation when n>2. In this paper we consider Moebius…

Perez proved some $L^2$ inequalities for closed convex hypersurfaces immersed in the Euclidean space $\mathbb{R}^{n+1}$, more generally, for closed hypersurfaces with non-negative Ricci curvature, immersed in an Einstein manifold. In this…