Pseudo-parallel (shortly PP) submanifolds are defined as a generalization of semi-parallel (shortly SP ) submanifolds and as extrinsic analogue of pseudo-symmetric (shortly PS) manifolds (in the sense of R. Deszcz) [1], [26]. Asperti et al [1] obtained a description of PP hypersurfaces in space form as quasiumbilic hypersurfaces or cyclids of Dupin and studied PP surfaces with maximal first nor...

We prove some Bernstein type theorems for entire space-like submanifolds in pseudo-Euclidean space and as a corollary, we give a new proof of the Calabi-Pogorelov theorem for Monge-Ampère equations.

The fundamental theorem of submanifolds is adapted to space-times. It is shown that the integrability conditions for the existence of submanifolds of a pseudo-Euclidean space contain the Einstein and Yang-Mills equations.

We establish the longtime existence and convergence results of the mean curvature flow of entire Lagrangian graphs in pseudo-Euclidean space.

The recovery of a set of points in an N -dimensional vector space from their set of mutual distances has a long history. In 1938 Young and Householder gave necessary and sufficient conditions for a set of numbers to be the set of mutual distances between a collection of real points in Euclidean space and gave methods for determining the configuration of these points up to translation and rotati...

A necessary algebraic condition is obtained for the intrinsic geometry of any spacetime locally and isometrically embedded into a pseudo-Euclidean 5-space.

‎The Lorentz transformation of order $(m=1,n)$‎, ‎$ninNb$‎, ‎is the well-known ‎Lorentz transformation of special relativity theory‎. ‎It is a transformation of time-space coordinates of the ‎pseudo-Euclidean space $Rb^{m=1,n}$ of one time dimension and ‎$n$ space dimensions ($n=3$ in physical applications)‎. ‎A Lorentz transformation without rotations is called a {it boost}‎. ‎Commonly‎, ‎the ...

Biharmonic surfaces in Euclidean space $mathbb{E}^3$ are firstly studied from a differential geometric point of view by Bang-Yen Chen, who showed that the only biharmonic surfaces are minimal ones. A surface $x : M^2rightarrowmathbb{E}^{3}$ is called biharmonic if $Delta^2x=0$, where $Delta$ is the Laplace operator of $M^2$. We study the $L_k$-biharmonic spacelike hypersurfaces in the $4$-dimen...

An isometric immersion f : Mn → M̃n from a Riemannian nmanifold Mn into a Kähler n-manifold M̃n is called Lagrangian if the complex structure J of the ambient manifold M̃n interchanges each tangent space of Mn with the corresponding normal space. In this paper, we completely classify spherical Lagrangian submanifolds in complex Euclidean spaces. In this paper, we also provide two corresponding cla...