Submanifolds In pseudo-Euclidean Spaces Research Articles

Classification of Rectifying Space-Like Submanifolds in Pseudo-Euclidean Spaces

Classification of Rectifying Space-Like Submanifolds in Pseudo-Euclidean Spaces

Biharmonic Submanifolds with Parallel Mean Curvature Vector in Pseudo-Euclidean Spaces

In this paper, we investigate biharmonic submanifolds in pseudo-Euclidean spaces with arbitrary index and dimension. We give a complete classification of biharmonic spacelike submanifolds with parallel mean curvature vector in pseudo-Euclidean spaces. We also determine all biharmonic Lorentzian surfaces with parallel mean curvature vector field in pseudo-Euclidean spaces.

Convex function on pseudo-Grassmann manifold and its applications for Bernstein-type theorem

We derive the Hessian estimate of the w-function defined on the pseudo-Grassmann manifold Gn,mm, which is convex by the estimate. As an application, we give a direct proof of the Bernstein-type theorem due to Y. Xin [Manuscripta Math., 2000, 103: 191–202]. We also estimate the squared norm of the second fundamental form of a complete spacelike submanifold in pseudo-Euclidean space in terms of the w-function and the mean curvature.

Mean Curvature Flow with Bounded Gauss Image

We study the mean curvature flow of a complete space-like submanifold in pseudo-Euclidean space with bounded Gauss image and bounded curvature. We establish a relevant maximum principle for our setting. Then, we can obtain the “confinable property” of the Gauss images and curvature estimates under the mean curvature flow. Thus we prove a corresponding long time existence result.

Realization of frobenius manifolds as submanifolds in pseudo-Euclidean spaces

We introduce a class of k-potential submanifolds in pseudo-Euclidean spaces and prove that for an arbitrary positive integer k and an arbitrary nonnegative integer p, each N-dimensional Frobenius manifold can always be locally realized as an N-dimensional k-potential submanifold in ((k + 1)N + p)-dimensional pseudo-Euclidean spaces of certain signatures. For k = 1 this construction was proposed by the present author in a previous paper (2006). The realization of concrete Frobenius manifolds is reduced to solving a consistent linear system of second-order partial differential equations.

Theory of submanifolds, associativity equations in 2D topological quantum field theories, and Frobenius manifolds

We prove that the associativity equations of two-dimensional topological quantum field theories are very natural reductions of the fundamental nonlinear equations of the theory of submanifolds in pseudo-Euclidean spaces and give a natural class of potential flat torsionless submanifolds. We show that all potential flat torsionless submanifolds in pseudo-Euclidean spaces bear natural structures of Frobenius algebras on their tangent spaces. These Frobenius structures are generated by the corresponding flat first fundamental form and the set of the second fundamental forms of the submanifolds (in fact, the structural constants are given by the set of the Weingarten operators of the submanifolds). We prove in this paper that each N-dimensional Frobenius manifold can locally be represented as a potential flat torsionless submanifold in a 2N-dimensional pseudo-Euclidean space. By our construction this submanifold is uniquely determined up to motions. Moreover, in this paper we consider a nonlinear system, which is a natural generalization of the associativity equations, namely, the system describing all flat torsionless submanifolds in pseudo-Euclidean spaces, and prove that this system is integrable by the inverse scattering method.

Structure of space-like submanifolds in pseudo-Euclidean space

Let M be an n-dimensional complete non-compact space-like submanifold in a pseudo-Euclidean Space R p n+ p of index p. In this paper, we find sufficient conditions for M to have only one end or finitely many ends.

Geometry of position functions of Riemannian submanifolds in pseudo-Euclidean space

It is well-known that the position function is the simplest and the most natural geometric object associated with a submanifold in a Euclidean space or, more generally, in a pseudo-Euclidean space. We call a submanifold T-constant (respectively, N-constant) if the tangential component (respectively, the normal component) of its position function has constant length. The main purpose of this paper is to classify space-like submanifolds in pseudo-Euclidean space which are either T-constant or N-constant.

Some Aspects of the global Geometry of Entire Space-Like Submanifolds

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

Parallel and semiparallel space-like surfaces in pseudo-Euclidean spaces

Parallel submanifolds in pseudo-Euclidean spaces are characterized locally by the system rh = 0. Submanifolds satisfying the integrability condition R h = 0 of this system are called semiparallel; geometrically they are 2nd-order envelopes of the parallel submanifolds. The existence and geometry of such two-dimensional Riemannian submanifolds (surfaces) are investigated and their complete classification is given. Moreover, it is shown that inE n withs > 0 do exist not totally geodesic minimal semiparallel space-like surfaces.

Submanifolds in pseudo-Euclidean spaces satisfying the condition Δx=Ax+B

In this paper we study pseudo-Riemannian submanifolds in ℝ n +k/t satisfying the condition Δx=Ax+B, whereA is an endomorphism of ℝ n +k/t andB is a constant vector in ℝ n +k/t . We give a characterization theorem whenA is a self-adjoint endomorphism. As for hypersurfaces we are able to obtain a classification theorem for any endomorphismA.

Finite type submanifolds in pseudo-Euclidean spaces and applications

By applying these results, we will provethe following. (1) There exist no compact space-like hypersurfaces with constantmean curvature and constant scalar curvature in the anti-de Sitter space-time;(2) Every compact hypersurface with constant mean curvature and constantscalar curvature in a hyperbolic space is a small hypersphere and (3) If