A rigidity theorem for a space-like graph¶of higher codimension

We prove a rigidity theorem for a space-like graph with parallel mean curvature of arbitrary dimension and codimension in pseudo-Euclidean space via properties of its harmonic Gauss map. We also give an estimate of the squared norm of the second fundamental form in terms of the mean curvature and the image diameter under the Gauss map for space-like submanifolds with parallel mean curvature in pseudo-Euclidean space. The estimate also implies the former theorem. Received: 10 December 1999     

Rigidity and mean curvature flow via harmonic Gauss maps

We investigate properties of harmonic Gauss maps and their applications to Lawson-Osserman’s problem, to the rigidity of space-like submanifolds in a pseudo-Euclidean space and to the mean curvature flow.     

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 flat torsionless potential submanifolds. We show that all flat torsionless potential 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 that each N-dimensional Frobenius manifold can be locally represented as a flat torsionless potential submanifold in a 2N-dimensional pseudo-Euclidean space. By our construction, this submanifold is uniquely determined up to motions. Moreover, we consider a nonlinear system that 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. To the memory of my wonderful mother Maya Nikolayevna Mokhova (4 May 1926–12 September 2006) Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 2, pp. 368–376, August, 2007.     

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.     

Nonlocal Hamiltonian operators of hydrodynamic type with flat metrics, integrable hierarchies, and the associativity equations

We solve the problem of describing all nonlocal Hamiltonian operators of hydrodynamic type with flat metrics. This problem is equivalent to describing all flat submanifolds with flat normal bundle in a pseudo-Euclidean space. We prove that every such Hamiltonian operator (or the corresponding submanifold) specifies a pencil of compatible Poisson brackets, generates bihamiltonian integrable hierarchies of hydrodynamic type, and also defines a family of integrals in involution. We prove that there is a natural special class of such Hamiltonian operators (submanifolds) exactly described by the associativity equations of two-dimensional topological quantum field theory (the Witten-Dijkgraaf-Verlinde-Verlinde and Dubrovin equations). We show that each N-dimensional Frobenius manifold can locally be represented by a special flat N-dimensional submanifold with flat normal bundle in a 2N-dimensional pseudo-Euclidean space. This submanifold is uniquely determined up to motions.     

Some results on space-like self-shrinkers

We study space-like self-shrinkers of dimension n in pseudo-Euclidean space R_m^m+n with index m. We derive drift Laplacian of the basic geometric quantities and obtain their volume estimates in pseudo-distance function. Finally, we prove rigidity results under minor growth conditions in terms of the mean curvature or the image of Gauss maps.     

AN ESTIMATE FOR THE MEAN CURVATURE OF SUBMANIFOLDS CONTAINED IN A HOROBALL

We obtain the Omori-Yau maximum principle on complete properly immersed submanifolds with the mean curvature satisfying certain condition in complete Riemannian manifolds whose radial sectional curvature satisfies some decay condition, which generalizes our previous results in [17]. Using this generalized maximum principle, we give an estimate on the mean curvature of properly immersed submanifolds in H^n × R^e with the image under the projection on H^n contained in a horoball and the corresponding situation in hyperbolic space. We also give other applications of the generalized maximum principle.     

The grassmannian manifolds with indefinite metrics

In this paper we study the geometrical properties of Grassmannian manifolds constructed in Minkowski space as submanifolds in a certain pseudo-Euclidean space and give a condition that the generalized Gauss map of a spacelike submanifold in Minkowski space is harmonic. This work is supported partially by the National Natural Science Foundation of China.     

On the total curvature of curves in a Minkowski space

We consider simple closed curves in a Minkowski space. We give bounds of the total Minkowski curvature of the curve in terms of the total Euclidean curvature and of normal curvatures on the indicatrix (supposed to be a central symmetric hypersurface) of the Minkowski norm. Corollaries of this result provide analogues to Fenchel and Fary-Milnor theorems. We also give an upper bound of the Minkowski length of a simple closed curve contained in a Minkowski ball of radius R, in terms of the total Minkowski curvature and of normal curvatures on the indicatrix. Whenever the Minkowski space is Euclidean our results reduce to the classical ones.     

Warped product structure of submanifolds with nonpositive extrinsic curvature in space forms

By means of a simple warped product construction we obtain examples of submanifolds with nonpositive extrinsic curvature and minimal index of relative nullity in any space form. We then use this to extend to arbitrary space forms four known splitting results for Euclidean submanifolds with nonpositive sectional curvature.     

Surfaces in a pseudo-Euclidean space

A survey of results on regular and nonregular surfaces in a three-dimensional pseudo-Euclidean space. The method of approximating a convex surface by polyhedra and the intrinsic construction of polyhedra of negative curvature are considered in detail. A theorem on the existence in a pseudo-Euclidean space of a convex polyhedron with given polyhedral metric of negative curvature with a finite number of vertices is proved.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 11, pp. 177–202, 1980.     

共形空间${\mathbb Q}^n_s$中的正则Blaschke拟全脐子流形

[Nie C X,Wu C X,Regular submanifolds in the conformal space Q_p~n,ChinAnn Math,2012,33B(5):695-714]中研究了共形空间Q_s~n中的正则子流形,并引入了共形空间Q_s~n中的子流形理论.本文作者将分类共形空间Q_s~n中的Blaschke拟全脐子流形,证明伪Riemann空间形式中具有常数量曲率和平行的平均曲率向量场的正则子流形是共形空间中的Blaschke拟全脐子流形;反之,共形空间中的Blaschke拟全脐子流形共形等价于伪Riemann空间形式中具有常数量曲率和平行的平均曲率向量场的正则子流形.这一结论可看作是共形空间Q_s~n中共形迷向子流形分类定理的推广.     

On special submanifolds of the Page space

In this paper, we study some classes of submanifolds of codimension one and two in the Page space. These submanifolds are totally geodesic. We also compute their curvature and show that some of them are constant curvature spaces. Finally, we give information on how the Page space is related to some other metrics on the same underlying smooth manifold.     

A note on biharmonic submanifolds of product spaces

We investigate biharmonic submanifolds of the product of two space forms. We prove a necessary and sufficient condition for biharmonic submanifolds in these product spaces. Then, we obtain mean curvature estimates for proper-biharmonic submanifold of a product of two unit spheres. We also prove a non-existence result in the case of the product of a sphere and a hyperbolic space.     

Complete space-like submanifolds in locally symmetric semi-definite spaces

The purpose of this paper is to study complete space-like submanifolds with parallel mean curvature vector and flat normal bundle in a locally symmetric semi-defnite space satisfying some curvature conditions. We first give an optimal estimate of the Laplacian of the squared norm of the second fundamental form for such submanifold. Furthermore, the totally umbilical submanifolds are characterized     

Curvature Estimates for Submanifolds in Warped Products

We give estimates of the intrinsic and the extrinsic curvature of manifolds that are isometrically immersed as cylindrically bounded submanifolds of warped products. We also address extensions of the results in the case of submanifolds of the total space of a Riemannian submersion.     

Singularities of Hyperbolic Gauss Maps

In this paper we adopt the hyperboloid in Minkowski space asthe model of hyperbolic space. We define the hyperbolic Gaussmap and the hyperbolic Gauss indicatrix of a hypersurface inhyperbolic space. The hyperbolic Gauss map has been introducedby Ch. Epstein [J. Reine Angew. Math. 372 (1986) 96–135]in the Poincaré ball model, which is very useful forthe study of constant mean curvature surfaces. However, it isvery hard to perform the calculation because it has an intrinsicform. Here, we give an extrinsic definition and we study thesingularities. In the study of the singularities of the hyperbolicGauss map (indicatrix), we find that the hyperbolic Gauss indicatrixis much easier to calculate. We introduce the notion of hyperbolicGauss–Kronecker curvature whose zero sets correspond tothe singular set of the hyperbolic Gauss map (indicatrix). Wealso develop a local differential geometry of hypersurfacesconcerning their contact with hyperhorospheres. 2000 MathematicalSubject Classification: 53A25, 53A05, 58C27.     

四元数射影空间的全实伪脐子流形

本文给出了四元数射影空间中紧致全实伪脐子流形关于截面曲率和Ricci曲率的Pinching定理，并推广和改进了四元数射影空间中紧致全实极小流形的一些结果．     

Chern-Lashof absolute curvature of complex submanifolds and volumes of Grassmann images

We introduce an analog of the Chern-Lashof absolute curvature for complex submanifolds in complex Euclidean spaces. A relation between this curvature and the volume of the Grassmann image of the submanifold is established.     

關閉撓曲線的全曲率

<正> §1.引言.大家知道,微分幾何學所討論的一般是關於圖形的局部性質.但是這些局部性質與圖形的整個性質間常存在有某些關係.討論圖形的整個性質的微分幾何學叫做整體性的.關於整體性微分幾何學有這樣的一個著名定理:設一關閉撓曲線C