非紧对称空间中子流形焦点和形算子间的关系

首先详细地讨论了非紧Ｌｉｅ群的度量和Ｃａｒｔａｎ分解,然后由Ｌｉｅ群和对称空间的关系得到了非紧对称空间中的子流形焦点存在的充要条件,同时还给出了焦点重数的计算方法．     

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     

球面上的极小子流形

本文证明了,在一定条件下,单位球面上具有任意余维数p的n维极小子流形是球面极小积上的一个开子流形,其中,从而推广了文[1]中命题1,那里余维数为1.     

A vanishing theorem for modular symbols on locally symmetric spaces

A modular symbol is the fundamental class of a totally geodesic submanifold embedded in a locally Riemannian symmetric space , which is defined by a subsymmetric space . In this paper, we consider the modular symbol defined by a semisimple symmetric pair (G,G'), and prove a vanishing theorem with respect to the -component in the Matsushima-Murakami formula based on the discretely decomposable theorem of the restriction . In particular, we determine explicitly the middle Hodge components of certain totally real modular symbols on the locally Hermitian symmetric spaces of type IV. Received: December 8, 1996     

Parallel submanifolds of complex projective space and their normal holonomy

The object of this article is to compute the holonomy group of the normal connection of complex parallel submanifolds of the complex projective space. We also give a new proof of the classification of complex parallel submanifolds by using a normal holonomy approach. Indeed, we explain how these submanifolds can be regarded as the unique complex orbits of the (projectivized) isotropy representation of an irreducible Hermitian symmetric space. Moreover, we show how these important submanifolds are related to other areas of mathematics and theoretical physics. Finally, we state a conjecture about the normal holonomy group of a complete and full complex submanifold of the complex projective space. Research partially supported by GNSAGA (INdAM) and MIUR of Italy.     

Submanifolds with Flat Normal Bundle

Let M be a compact orientable submanifold immersed in a Riemannian manifold of constant curvature with flat normal bundle. This paper gives intrinsic conditions for M to be totally umbilical or a local product of several totally umbilical submanifolds. It is proved especially that a compact hypersurface in the Euclidean space with constant scalar curvature and nonnegative Ricci curvature is a sphere.     

Enlarging totally geodesic submanifolds of symmetric spaces to minimal submanifolds of one dimension higher

We show that a totally geodesic submanifold of a symmetric space satisfying certain conditions admits an extension to a minimal submanifold of dimension one higher, and we apply this result to construct new examples of complete embedded minimal submanifolds in simply connected noncompact globally symmetric spaces.

     

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.

     

ON THE INVARIANT SUBMANIFOLDS OF RIEMANNIAN PRODUCT MANIFOLD

In this paper, the vertical and horizontal distributions of an invariant submanifold of a Riemannian product manifold are discussed. An invariant real space form in a Riemannian product manifold is researched. Finally, necessary and sufficient conditions are given on an invariant submanifold of a Riemannian product manifold to be a locally symmetric and real space form.     

Affine Locally Symmetric Surfaces in R~4

The nondegenerate affine locally symmetric surfaces in R^4 with the transversal bundle defined by Nomizu and Vrancken have been studied and a complete classification of the locally symmetric surfaces with flat normal bundle has been given.     

On the realization of Riemannian symmetric spaces in Lie groups II

In this paper we generalize a result in [J. An, Z. Wang, On the realization of Riemannian symmetric spaces in Lie groups, Topology Appl. 153 (7) (2005) 1008-1015, showing that an arbitrary Riemannian symmetric space can be realized as a closed submanifold of a covering group of the Lie group defining the symmetric space. Some properties of the subgroups of fixed points of involutions are also proved.     

Clifford systems, algebraically constant second fundamental form and isoparametric hypersurfaces

In this paper we prove that a submanifold with parallel mean curvature of a space of constant curvature, whose second fundamental form has the same algebraic type as the one of a symmetric submanifold, is locally symmetric. As an application, using properties of Clifford systems, we give a short and alternative proof of a result of Cartan asserting that a compact isoparametric hypersurface of the sphere with three distinct principal curvatures is a tube around the Veronese embedding of the real, complex, quaternionic or Cayley projective planes. Received: 22 April 1998     

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.     

Screen conformal lightlike submanifolds of semi-Riemannian manifolds

Since the induced objects on a lightlike submanifold depend on its screen distribution which, in general, is not unique and hence we can not use the classical submanifold theory on a lightlike submanifold in the usual way. Therefore, in present paper, we study screen conformal lightlike submanifolds of a semi-Riemannian manifold, which are essential for the existence of unique screen distribution. We obtain a characterization theorem for the existence of screen conformal lightlike submanifolds of a semi-Riemannian manifold. We prove that if the differential operator D^s is a metric Otsuki connection on transversal lightlike bundle for a screen conformal lightlike submanifold then semi-Riemannian manifold is a semi-Euclidean space. We also obtain some characterization theorems for a screen conformal totally umbilical lightlike submanifold of a semi-Riemannian space form. Further, we obtain a necessary and sufficient condition for a screen conformal lightlike submanifold of constant curvature to be a semi-Euclidean space. Finally, we prove that for an irrotational screen conformal lightlike submanifold of a semi-Riemannian space form, the induced Ricci tensor is symmetric and the null sectional curvature vanishes.     

Curvature of the space of positive Lagrangians

A Lagrangian submanifold in an almost Calabi–Yau manifold is called positive if the real part of the holomorphic volume form restricted to it is positive. An exact isotopy class of positive Lagrangian submanifolds admits a natural Riemannian metric. We compute the Riemann curvature of this metric and show all sectional curvatures are non-positive. The motivation for our calculation comes from mirror symmetry. Roughly speaking, an exact isotopy class of positive Lagrangians corresponds under mirror symmetry to the space of Hermitian metrics on a holomorphic vector bundle. The latter space is an infinite-dimensional analog of the non-compact symmetric space dual to the unitary group, and thus has non-positive curvature.     

de Sitter空间中法联络平坦的完备类空子流形

该文证明了de Sitter空间中具有平行平均曲率向量的常数量曲率完备类空子流形,如果其法联络是平坦的,且M的截面曲率小于0,或M的第二基本形式模长平方‖σ‖相似文献   

关于四元射影空间的一般极小子流形

本文研究四元射影空间的一般极小子流形,建立了一个积分公式,并且用此公式得到了紧致一般极小子流形的截曲率的Ｐｉｎｃｈｉｎｇ结果．     

On the Geometry of the Orbits of Hermann Actions

We investigate the submanifold geometry of the orbits of Hermann actions on Riemannian symmetric spaces. After proving that the curvature and shape operators of these orbits commute, we calculate the eigenvalues of the shape operators in terms of the restricted roots. As applications, we get a formula for the volumes of the orbits and a new proof of a Weyl-type integration formula for Hermann actions.      

An Expression for the Riemann Tensor of a Submanifold Given by a System of Equations in a Riemannian Space

We derive an expression for the Riemann tensor of a submanifold given implicitly by a system of independent equations in a Riemannian space. In particular, we prove a formula for the internal curvature of a two-surface in a three-dimensional Riemannian space. Some applications of the formula are given.     

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.