Geometry of the Second-Order Tangent Bundles of Riemannian Manifolds

Let (M,g) be an n-dimensional Riemannian manifold and T2M be its secondorder tangent bundle equipped with a lift metric (g).In this paper,first,the authors construct some Riemannian almost product structures on (T2M,(g)) and present some results concerning these structures.Then,they investigate the curvature properties of (T2M,(g)).Finally,they study the properties of two metric connections with nonvanishing torsion on (T2 M,(g)):The H-lift of the Levi-Civita connection of g to T2 M,and the product conjugate connection defined by the Levi-Civita connection of (g) and an almost product structure.     

On the Tangent Bundle of a Hypersurface in a Riemannian Manifold

Let(Mn, g) and(Nn+1, G) be Riemannian manifolds. Let TMn and TNn+1 be the associated tangent bundles. Let f :(Mn,g) →(Nn+1,G) be an isometrical immersion with g = f*G, F =(f, df) :(TMn, ■) →(TNn+1, Gs) be the isometrical immersion with ■= F*Gs where (df)x: TxM → Tf(x)N for any x ∈M is the differential map, and Gs be the Sasaki metric on TN induced from G. This paper deals with the geometry of TMn as a submanifold of TNn+1 by the moving frame method. The authors firstly study the extrinsic geometry of TMn in TNn+1. Then the integrability of the induced almost complex structure of TM is discussed.     

Notes on the Rescaled Sasaki Type Metric on the Cotangent Bundle

Let (M,g) be an n-dimensional Riemannian manifold and T^*M be its cotangent bundle equipped with the rescaled Sasaki type metric. In this paper, we firstly study the paraholomorphy property of the rescaled Sasaki type metric by using some compatible paracomplex structures on T^*M. Second, we construct locally decomposable Golden Riemannian structures on T^*M. Finally we investigate curvature properties of T^*M.     

局部共形对称黎曼流形上的保圆曲率张量长度的整体刚性定理

本文我们研究了局部共形对称闭黎曼流形, 建立了一个关于保圆曲率矢量长度的整体刚度定理.     

Some Aspects on the Geometry of the Tangent Bundles and Tangent Sphere Bundles of a Riemannian Manifold

In this paper we study a Riemannian metric on the tangent bundle T(M) of a Riemannian manifold M which generalizes Sasaki metric and Cheeger–Gromoll metric and a compatible almost complex structure which confers a structure of locally conformal almost K?hlerian manifold to T(M) together with the metric. This is the natural generalization of the well known almost K?hlerian structure on T(M). We found conditions under which T(M) is almost K?hlerian, locally conformal K?hlerian or K?hlerian or when T(M) has constant sectional curvature or constant scalar curvature. Then we will restrict to the unit tangent bundle and we find an isometry with the tangent sphere bundle (not necessary unitary) endowed with the restriction of the Sasaki metric from T(M). Moreover, we found that this map preserves also the natural contact structures obtained from the almost Hermitian ambient structures on the unit tangent bundle and the tangent sphere bundle, respectively. This work was also partially supported by Grant CEEX 5883/2006–2008, ANCS, Romania.     

Kahler流形上的Lagrange向量场*

本文中。我们讨论K?hler流形上的Lagrange向量场,并用它来描述和解决Khler流形上的Newton力学和Lagrange力学中的一些问题。     

The Cauchy problem about Dirac-wave map from the 2-dimension Minkowski space to a complete Riemannian manifold

When a target manifold is complete with a bounded curvature, we prove that there exists a unique global solution which satisfies the Euler-lagrange equation of for the given Cauchy data.     

过三次曲线上一点的切线的求法

通过对一道求切线例题的讨论,引申出求过一条三次曲线上一点的切线的求法,进而推出求解这类问题的一个一般性结论.     

Stein流形上全纯函数积分公式的拓广

得到了Stein流形上一种全纯函数的积分式，这种公式的特点含有可供选择的参数m≥2的整数，当m=2时即为Stein流形上全纯函数的方B-M公式．     

Prescribing Curvature Problems on the Bakry-Emery Ricci Tensor of a Compact Manifold with Boundary

t The authors consider the problem of conformally deforming a metric such that the k-curvature defined by an elementary symmetric function of the eigenvalues of the Bakry-Emery Ricci tensor on a compact manifold with boundary to a prescribed function. A consequence of our main result is that there exists a complete metric such that the Monge-Amp~re type equation with respect to its Bakry-Emery Ricci tensor is solvable, provided that the initial Bakry-Emery Ricci tensor belongs to a negative convex cone.     

Natural <Emphasis Type="Italic">T</Emphasis>-Functions on the Cotangent Bundle of a Weil Bundle

A natural T-function on a natural bundle F is a natural operator transforming vector fields on a manifold M into functions on FM. For any Weil algebra A satisfying dim M width(A) + 1 we determine all natural T-functions on T ^* T ^A M, the cotangent bundle to a Weil bundle T ^A M.     

On the Geometry of the （1, 1）-Tensor Bundle with Sasaki Type Metric

Curvature properties are studied for the Sasaki metric on the (1,1) tensor bundle of a Riemannian manifold. As an application, examples of almost para-Nordenian and para-K(a)hler-Nordenian B-metrics are constructed on the (1,1) tensor bundle by looking at the Sasaki metric. Also, with respect to the para-Nordenian B-structure, paraholomorphic conditions for the complete lifts of vector fields are analyzed.     

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.     

A Note on the Infimum of Energy of Unit Vector Fields on a Compact Riemannian Manifold

Our main result in this paper establishes that if G is a compact Lie subgroup of the isometry group of a compact Riemannian manifold M acting with cohomogeneity one in M and either G has no singular orbits or the singular orbits of G have dimension at most n−3, then the unit vector field N orthogonal to the principal orbits of G is weakly smooth and is a critical point of the energy functional acting on the unit normal vector fields of M. A formula for the energy of N in terms of the of integral of the Ricci curvature of M and of the integral of the square of the mean curvature of the principal orbits of G is obtained as well. In the case that M is the sphere and G the orthogonal group it is known that that N is minimizer. It is an open question if N is a minimizer in general.     

A Girsanov Type Theorem on the Path Space Over a Compact Riemannian Manifold

Abstract

This article establishes a Girsanov type theorem on path spaces over compact Riemannian manifolds, generalizing the classical Girsanov theorem for Euclidean spaces.     

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   

Complete Formulas for the Volumes of Tubes about Curves in a Riemannian Manifold

By using the Taylor expansions of the solutions of Jacobi equations, we obtain the complete formulas for the volumes of tubes about curves in a Riemannian manifold. This unifies the known results and simplifies the computations involved in this direction. In the special case of surfaces, we also obtain the corresponding complete formulas which generalize the known results. Received July 23, 1998, Accepted January 14, 1999     

On Affine Connections Induced on the (1,1)-Tensor Bundle

Let M be an n-dimensional differentiable manifold with an affine connection without torsion and T_1~1(M) its(1, 1)-tensor bundle. In this paper, the authors define a new affine connection on T_1~1(M) called the intermediate lift connection, which lies somewhere between the complete lift connection and horizontal lift connection. Properties of this intermediate lift connection are studied. Finally, they consider an affine connection induced from this intermediate lift connection on a cross-section σ_ξ(M) of T_1~1(M) defined by a(1, 1)-tensor field ξ and present some of its properties.