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.     

A BINARY INFINITESIMAL FORM OF TEICHMLLER METRIC AND ANGLES IN AN ASYMPTOTIC TEICHMLLER SPACE

The geometry of Teichmller metric in an asymptotic Teichmller space is studied in this article. First, a binary infinitesimal form of Teichmller metric on AT(X) is proved.Then, the notion of angles between two geodesic curves in the asymptotic Teichmller space AT(X) is introduced. The existence of such angles is proved and the explicit formula is obtained. As an application, a sufficient condition for non-uniqueness geodesics in AT(X) is obtained.     

GENERALIZED LERAY FORMULA ON POSITIVE COMPLEX LAGRANGE-GRASSMANN MANIFOLDS

A full proof of a matrix lemma stated in[1]is given,and the notions concerningcannonical argument and signature of a triple of the Lagrange planes in a complex phasespace is formulated.Then a formula is established,which generalizes that one of J.Leray'sin real phase space case.Finally,some applications of the formula are given.     

The Approach to the Covariance of the Outer Producter of Two Independent Random Vector in Inter Productor Space

The purpose of this article is to prescnt by using vector space methods. a formula as how to calculate the covariance of the outer product of two independent random vecters in inter product space and to makes a discussion on the covariance of the orthogonally invariant random vector and that of the weakly spherically distributed orter produt.     

THE CODIMENSION FORMULA ON QUASI-INVARIANT SUBSPACES OF THE FOCK SPACE

Let M be an approximately finite codimensional quasi-invariant subspace of the Fock space. This paper gives a formula to calculate the codimension of such spaces and uses this formula to study the structure of quasi-invariant subspaces of the Fock space. Especially, as one of applications, it is showed that the analogue of Beurling's theorem is not true for the Fock space L_a~2 in the case of n > 2.     

Complete Totally Real Submanifolds with Parallel Mean Curvature

i. Introduction A submanifold M in a Kaehler manifold M is said to be totally real if every tangent space of M is mapped into its normal space by the complex structure of M. Some fundamental properties of totally real submanifolds can be found in [1], [2]. Let σ be the second fundamental form of M. The mean curvature η of M is defined by η=tr σ , and M is called a submanifold with     

A Note on Residue Formulas for the Euler Class of Sphere Fibrations

This paper presents a definition of residue formulas for the Euler class of cohomology-oriented sphere fibrations ξ. If the base of ξ is a topological manifold, a Hopf index theorem can be obtained and, for the smooth category, a generalization of a residue formula is derived for real vector bundles given in [2].     

关于复射影空间中具有平行平均曲率向量的全实叶的注记(英文)

We discussed a totally real Riemannian foliations with parallel mean curvature on a complex projective space.We carried out the divergence of a vector field on it and obtained a formula of Simons’type.     

A complete classification of Blaschke parallel submanifolds with vanishing Mbius form

The Blaschke tensor and the Mbius form are two of the fundamental invariants in the Mobius geometry of submanifolds;an umbilic-free immersed submanifold in real space forms is called Blaschke parallel if its Blaschke tensor is parallel.We prove a theorem which,together with the known classification result for Mobius isotropic submanifolds,successfully establishes a complete classification of the Blaschke parallel submanifolds in S~n with vanishing Mobius form.Before doing so,a broad class of new examples of general codimensions is explicitly constructed.     

Discrete chaos in Banach spaces

This paper is concerned with chaos in discrete dynamical systems governed by continuously Frech@t differentiable maps in Banach spaces. A criterion of chaos induced by a regular nondegenerate homoclinic orbit is established. Chaos of discrete dynamical systems in the n-dimensional real space is also discussed, with two criteria derived for chaos induced by nondegenerate snap-back repellers, one of which is a modified version of Marotto's theorem. In particular, a necessary and sufficient condition is obtained for an expanding fixed point of a differentiate map in a general Banach space and in an n-dimensional real space, respectively. It completely solves a long-standing puzzle about the relationship between the expansion of a continuously differentiable map near a fixed point in an n-dimensional real space and the eigenvalues of the Jacobi matrix of the map at the fixed point.     

Kinematic measure of segments lying in a domain

The paper proves a formula for calculation of the kinematic measure K(D, l) of set of segments with constant length l, entirely contained in a bounded convex domainDof the Euclidean space. The obtained formula permits to find an explicit form for the kinematic measure K(D, l) for the domains D with known chord length distribution. In particular, application of the obtained formula gives explicit expressions for K(D, l) in the disc, regular triangle, rectangle and regular pentagon.     

Sasakian空间形式中具有平行平均曲率向量的C-全实子流形

本文讨论了Sasakian空间形式中具有平行平均曲率向量的C-全实子流形,得到了一个Simons型公式并且改进了S.Yamaguchi等的一个结果.     

Integral geometry of complex space forms

We show how Alesker’s theory of valuations on manifolds gives rise to an algebraic picture of the integral geometry of any Riemannian isotropic space. We then apply this method to give a thorough account of the integral geometry of the complex space forms, i.e. complex projective space, complex hyperbolic space and complex Euclidean space. In particular, we compute the family of kinematic formulas for invariant valuations and invariant curvature measures in these spaces. In addition to new and more efficient framings of the tube formulas of Gray and the kinematic formulas of Shifrin, this approach yields a new formula expressing the volumes of the tubes about a totally real submanifold in terms of its intrinsic Riemannian structure. We also show by direct calculation that the Lipschitz-Killing valuations stabilize the subspace of invariant angular curvature measures, suggesting the possibility that a similar phenomenon holds for all Riemannian manifolds. We conclude with a number of open questions and conjectures.     

Kinematische Berührung im Äquiaffinen

Given two curves in the real affine plane, one is fixed and the other undergoes volume-preserving affinities. Through transversal affinities we define a contact measure on the subset consisting of those affinities, which cause third-order contact between the fixed and the transformed curve. A kinematic formula expresses this contact measure in terms of affine lengths and affine curvatures of the given curves. In a similar way, parallel supporting planes of closed convex surfaces in affine space are treated.     

Parallel mean curvature biconservative surfaces in complex space forms

In this paper, we consider PMC surfaces in complex space forms, and study the interaction between the notions of PMC, totally real and biconservative. We first consider PMC surfaces in a non-flat complex space form and prove that they are biconservative if and only if totally real. Then, we find a Simons-type formula for a well-chosen vector field constructed from the mean curvature vector field and use it to prove a rigidity result for CMC biconservative surfaces in two-dimensional complex space forms. We prove then a reduction codimension result for PMC biconservative surfaces in non-flat complex space forms. We conclude by constructing examples of CMC non-PMC biconservative submanifolds from the Segre embedding and discuss when they are proper-biharmonic.     

Explicit solution of the inverse kinematic problem in the non-Herglotz case

The inverse kinematic problem is solved in the half space R ₊ ^ν+1 ={(x,z)|z?0,x∈R^ν, ν?1 under the assumption that the index of refraction can be represented in the form $$n^2 (x,z) = K^2 (z) + \sum\limits_{j = 1}^\nu {\Phi _j^2 (x_j ),} n_z< 0.$$ . The solution obtained is a generalization of the Herglotz-Wiechert formula. A formula is presented for the solution of the inverse kinematic problem in the general case of separation of variables in the eikonal equation.     

Half Space and Quarter Space Dirichlet Problems for the Partial Differential Equation

A customary, heuristic, method, by which the Poisson integral formula for the Dirichlet problem, for the half space, for Laplace's equation is obtained, involves Green's function, and Kelvin's method of images. Although this heuristic method leads one to guess the correct result, this Poisson formula still has to be verified directly, independently of the method by which it was arrived at, in order to be absolutely certain that a solution of the Dirichlet problem for the half space, for Laplace's equation, has been actually obtained. A similar heuristic method, as seems to be generally known, could be followed in solving the Dirichlet problem, for the half space, for the equation where is a real constant. However, in Part 1, a different, labor-saving, method is used to study Dirichlet problems for the equation. This method is essentially based on what Hadamard called the method of descent. Indeed, it is shown that he who has solved the half space Dirichlet problem for Laplace's equation has already solved the half space Dirichlet problem for the equation In Part 2, the solution formula for the quarter space Dirichlet problem for Laplace's equation is obtained from the Poisson integral formula for the half space Dirichlet problem for Laplace's equation. A representation theorem for harmonic functions in the quarter space is deduced. The method of descent is used, in Part 3, to obtain the solution formula for the quarter space Dirichlet problem for the equation by means of the solution formula for the quarter space Dirichlet problem for Laplace's equation. So that, indeed, it is also shown that he who has solved the quarter space Dirichlet problem for Laplace's equation has already solved the quarter space Dirichlet problem for the " equation" For the sake of completeness and clarity, and for the convenience of the reader, the appendix, at the end of Part 3, contains a detailed proof that the Poisson integral formula solves the half space Dirichlet problem for Laplace's equation. The Bibliography for Parts 1,2, 3 is to be found at the end of Part 1.     

Critical curves for the total normal curvature in surfaces of 3-dimensional space forms

A variational problem closely related to the bending energy of curves contained in surfaces of real 3-dimensional space forms is considered. We seek curves in a surface which are critical for the total normal curvature energy (and its generalizations). We start by deriving the first variation formula and the corresponding Euler–Lagrange equations of these energies and apply them to study critical special curves (geodesics, asymptotic lines, lines of curvature) on surfaces. Then, we show that a rotation surface in a real space form for which every parallel is a critical curve must be a special type of a linear Weingarten surface. Finally, we give some classification and existence results for this family of rotation surfaces.     

Sasakian空间型的C-全实伪脐子流形

本文讨论了Ｓａｓａｋｉａｎ空间型的Ｃ-全实伪脐子流形，给出关于第二基本形式长度的一个Ｐｉｎｃｈｉｎｇ定理．     

On Sampling Formulas on Symmetric Spaces

We give a sampling formula using the Radon transform along a maximal geodesic subspace of the Riemannian symmetric space. For the real hyperbolic space we can get a total sampling formula. To get this formula, we prepare a sampling formula for the sphere.