Some remarks on pseudo-umbilical totally real submanifolds in a complex projective space

This paper studies the relationship between the pseudo-umbilical totally real submanifolds and the minimal totally real submanifolds in a complex projective space. Two theo- rems which claim that some types of pseudo-umbilical totally real submanifolds must be minimal submanifolds are proved.     

The constancy of principal curvatures of curvature-adapted submanifolds in symmetric spaces

In this paper, we investigate complete curvature-adapted submanifolds with maximal flat section and trivial normal holonomy group in symmetric spaces of compact type or non-compact type under a certain condition, and derive the constancy of the principal curvatures of such submanifolds. As a result, we derive that such submanifolds are isoparametric.     

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.     

常曲率空间中的具有共形第二基本形式的子流形

以把调和态射看作等距浸入的单位法投影的问题为背景,研究了具有共形第二基本形式的子流形,论证了具有共形第二基本形式的高维子流形,一般不是由极小点和全脐点构成,这和曲面的情形形成了鲜明的对照。也给出了常曲率空间中具有平行中曲率的奇数维子流形的一个完全分类。     

Graham-Witten's Conformal Invariant for Closed Four Dimensional Submanifolds

It was proved by Graham and Witten in 1999 that conformal invariants of submanifolds can be obtained via volume renormalization of minimal surfaces in conformally compact Einstein manifolds. The conformal invariant of a submanifold $\Sigma$ is contained in the volume expansion of the minimal surface which is asymptotic to $\Sigma$ when the minimal surface approaches the conformaly infinity. In the paper we give the explicit expression of Graham-Witten's conformal invariant for closed four dimensional submanifolds and find critical points of the conformal invariant in the case of Euclidean ambient spaces.     

Construction of Lagrangian Submanifolds in Complex Hyperquadric

In this paper, the authors present a method to construct the minimal and ${\rm H}$-minimal Lagrangian submanifolds in complex hyperquadric $Q_n$ from submanifolds with special properties in odd-dimensional spheres. The authors also provide some detailed examples.     

On second order minimal Lagrangian sub manifolds in complex space forms

In this paper, we determine all second order minimal Lagrangian submanifolds in complex space forms whose cubic forms have the largest non-trivial continuous symmetries. We describe these minimal Lagrangian submanifolds from the viewpoint of Bryant and study their geometric properties.     

New construction of special Lagrangian submanifolds in T^＊R^n equipped with the standard metric

A class of twisted special Lagrangian submanifolds in T~*R~n and a kind of austere submanifold from conormal bundle of minimal surface of R~3 are constructed.     

New construction of special Lagrangian submanifolds in <Emphasis Type="Italic">T</Emphasis>*R<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> equipped with the standard metric

A class of twisted special Lagrangian submanifolds in T＊R^n and a kind of austere submanifold from conormal bundle of minimal surface of R^3 are constructed.     

A criterion for reduction of variables in the Willmore-Chen variational problem and its applications

We exhibit a criterion for a reduction of variables for Willmore-Chen submanifolds in conformal classes associated with generalized Kaluza-Klein metrics on flat principal fibre bundles. Our method relates the variational problem of Willmore-Chen with an elasticity functional defined for closed curves in the base space. The main ideas involve the extrinsic conformal invariance of the Willmore-Chen functional, the large symmetry group of generalized Kaluza-Klein metrics and the principle of symmetric criticality. We also obtain interesting families of elasticae in both lens spaces and surfaces of revolution (Riemannian and Lorentzian). We give applications to the construction of explicit examples of isolated Willmore-Chen submanifolds, discrete families of Willmore-Chen submanifolds and foliations whose leaves are Willmore-Chen submanifolds.

     

The Conjugate Nets, Cartan Submanifolds, and Laplace Transformations in Space Forms

In this paper we establish the geometric theory of conjugate nets, Cartan submanifolds, and Laplace transformations in sphere and pseudo-sphere spaces. The corresponding theory in cases of projective and Euclidean spaces has been established by Chern, Kamran and Tenenblat.     

Biharmonic Submanifolds in 0-Pinched Riemannian Manifolds

We study biharmonic submanifolds in &pinched Riemannian manifolds,and obtain some sufficient conditions for biharmonic submanifolds to be minimal ones.     

Locally symmetric minimal affine Lagrangian surfaces in C<Superscript>2</Superscript>

One of the basic facts known in the theory of minimal Lagrangian surfaces is that a minimal Lagrangian surface of constant curvature in C ² must be totally geodesic. In affine geometry the constancy of curvature corresponds to the local symmetry of a connection. In Opozda (Geom. Dedic. 121:155–166, 2006), we proposed an affine version of the theory of minimal Lagrangian submanifolds. In this paper we give a local classification of locally symmetric minimal affine Lagrangian surfaces in C ². Only very few of surfaces obtained in the classification theorems are Lagrangian in the sense of metric (pseudo-Riemannian) geometry. The research supported by the KBN grant 1 PO3A 034 26.     

黎曼流形在正交联络下的全脐点子流形

本文研究了正交联络下子流形基本方程以及在全脐点子流形中的应用.利用Cartan的方法将挠率张量分解成三个部分,计算得到正交联络下的三个基本方程,并考虑一个特殊的正交联络,证明了其黎曼曲率会有类似于Levi-Civita联络下的性质.利用基本方程得到常曲率空间中的全脐点子流形的性质,推广了Levi-Civita联络下的相应结果.     

紧致秩1对称空间子流形的总平均曲率

使用Ｒｉｅｍａｎｎ侵没的方法,得到了关于紧致秩１对称空间中子汉形的总平均曲率的积分不等式。     

Totally geodesic submanifolds of maximal rank in symmetric spaces

We gave a complete list of totally geodesic submanifolds of maximal rank in symmetric spaces of noncompact type. The compact cases can be obtained by the duality.     

Scalar curvature of a class of minimal submanifolds in rank 1 symmetric spaces

In this paper, we give some rigidity theorems which concern with compact minimal coisotropic submanifolds in ℂPⁿ, compact minimal quaternionic coisotropic submanifolds in ℚPⁿ and compact minimal hypersurfaces in P² (Cay).      

$\delta$-拼挤流形中的2-调和子流形

We study biharmonic submanifolds in δ-pinched Riemannian manifolds, and obtain some sufficient conditions for biharmonic submanifolds to be minimal ones.     

Semi-Slant Submanifolds of a Sasakian Manifold

We define and study both bi-slant and semi-slant submanifolds of an almost contact metric manifold and, in particular, of a Sasakian manifold. We prove a characterization theorem for semi-slant submanifolds and we obtain integrability conditions for the distributions which are involved in the definition of such submanifolds. We also study an interesting particular class of semi-slant submanifolds.     

A splitting theorem for equifocal submanifolds in simply connected compact symmetric spaces

A submanifold in a symmetric space is called equifocal if it has a globally flat abelian normal bundle and its focal data is invariant under normal parallel transportation. This is a generalization of the notion of isoparametric submanifolds in Euclidean spaces. To each equifocal submanifold, we can associate a Coxeter group, which is determined by the focal data at one point. In this paper we prove that an equifocal submanifold in a simply connected compact symmetric space is a non-trivial product of two such submanifolds if and only if its associated Coxeter group is decomposable. As a consequence, we get a similar splitting result for hyperpolar group actions on compact symmetric spaces. These results are an application of a splitting theorem for isoparametric submanifolds in Hilbert spaces by Heintze and Liu.