Differential geometry of submanifolds with planar normal sections

Summary Submanifolds of E^m with (pointwise) planar normal sections were studied in [3, 4]. In the present paper, we will prove that the first normal spaces of such submanifolds are always parallel in the normal bundle. Moreover, we will prove that a spherical submanifold has parallel second fundamental form if and only if it has pointwise planar normal sections. Some applications of these fundamental results will also be given.     

Classification of surfaces with pointwise planar normal sections and its application to Fomenko's conjecture

In this paper, we classify surfaces in E^m with pointwise planar normal sections. Our classification solves completely an open problem proposed in [4]. In particular, our result gives a solution to the Fomenko conjecture [6] concerning surfaces with zero normal torsion in E⁴.     

Semi-parallel surfaces in Euclidean space

Semi-parallel immersions are defined as extrinsic analogue for semi-symmetric spaces and as a direct generalization of parallel immersions. Using results of Backes on Euclidean Jordan triple systems, the totally geodesic immersions are shown to be the only minimal semi-parallel immersions into a Euclidean space. Semi-parallel immersions of surfaces into E^m are studied and a classification of semi-parallel immersions with pointwise planar normal sections of surfaces in E^m is given.Research Assistant of the National Fund of Scientific Research     

Classification of surfaces with planar normal sections

A surface in E^m is said to have planar normal sections if normal sections of M are planar curves. In this paper we completely classify surfaces in E^m with planar normal sections. Consequently, a new characterization of the Veronese surface is obtained.     

Surfaces in a pseudo-euclidean space with planar normal sections

A pseudo-Riemannian surface M in a pseudo-Euclidean space E _s ^m is said to have planar normal sections if normal sections of M are planar curves. In the present paper we give some classification theorem concerning surfaces in E _m ^s with planar normal sections.     

Affine planar geodesic immersions with maximal codimension

This work gives a classification theorem for affine immersions with planar geodesics in the case where the codimension is maximal. Vrancken classified parallel affine immersions in this case and obtained, among others, generalized Veronese submanifolds. In this work it is shown that the immersions with planar geodesics are the same as the parallel ones in the considered case. A geometric interpretation of parallel immersions is also given： The affine immersions with pointwise planar normal sections （with respect to the equiaffine transversal bundle） are parallel. This result is verified for surfaces in R4 and for immersions with the maximal codimension.     

On non–symmetric generalized schrodinger semigroup

In this paper some general phenomena are described for not necessarily systemeric so–called generalized Schrödinger semigroups (or generalized absorption/exciatation semigroups). These results are also applicable in case we consider Schrödinger semigroups on R ^v. In particular we describe some results on integral kernels: continuity, pointwise inequalities, ultracontractivity etc.For these inequalities we use a kind of stochastic bridge measure. The operator H is a closed linear extension of the operator H ⁰ + V in the space C ⁰(E) Here E is a locally compact second countable Hausdorff space and –H ₀ is supposed to generate a Feller semigroup in C ₀(E). Results in L^p (E,m) are also availale. Some examples are given     

Products and factors of Banach function spaces

Given two Banach function spaces we study the pointwise product space E · F, especially for the case that the pointwise product of their unit balls is again convex. We then give conditions on when the pointwise product E · M(E, F) = F, where M(E, F) denotes the space of multiplication operators from E into F.     

On isometric immersions with flat normal connection of the hyperbolic space L n into Euclidean space E n+m

We prove that the hyperbolic space L ⁿ cannot be immersed in an Euclidean space E ^n+m with a flat normal connection provided the module of the mean curvature vector is bounded.     

An extension of Takahashi's theorem

A classical result of T. Takahashi [8] is generalized to the case of hypersurfaces in the Euclidean space E ^m . More concretely, we classify Euclidean hypersurfaces whose coordinate functions in E ^m are eigenfunctions of their Laplacian.Partially supported by a CAICYT Grant PR84-1242-C02-02 Spain.     

Immersions with circular normal sections and normal sections of product immersions

We study immersions with normal sections that are circles in the ambient Euclidean space and formulate lemmas concerning normal sections of product immersions. As applications we determine all parallel immersions with planar normal sections and all immersions with planar normal sections and trivial normal connection.Aspirant N.F.W.O.     

Construction of markov process with markov resolvent onL P (E, m)

LetE be a Hausdorff topological space and let β(E) be its Borel σ-field. Letm be a σ-finite measure on (E, β(E)). A necessary and sufficient condition for a Markov resolvent onL ^P (E,m) to be associated with anm-tightm-special standard process (with state spaceE) is given. Furthermore some new examples which do not belong to the framework of Dirichlet space are also given. Project supported by the Youth Foundation of Chinese Academy of Sciences.     

The number of nearest and farthest points to a compactum in Euclidean space

A typical (in the sense of Baire category) compactA inE, whereE is either the Euclidean spaceE ⁸,s≧2, or the separable Hilbert space ℍ, generates a dense subsetC ^n,m(A) of the underlying space, such that everyx∈C ^n,m(A) has exactlyn nearest andm farthest points fromA, whenevern andm are positive integers satisfyingn+m≦ dimE+2. Research of this author is in part supported by Consiglio Nazionale delle Ricerche, G.N.A.F.A., Italy.     

On extension of isometries and approximate isometries between unit spheres

In this paper, we will study the isometric extension problem for L¹-spaces and prove that every surjective isometry from the unit sphere of L¹(μ) onto that of a Banach space E can be extended to a linear surjective isometry from L¹(μ) onto E. Moreover, we introduce the approximate isometric extension problem and show that, if E and F are Banach spaces and E satisfies the property (m) (special cases are L^∞(Γ), C₀(Ω) and L^∞(μ)), then every bijective ?-isometry between the unit spheres of E and F can be extended to a bijective 5?-isometry between their closed unit balls. At last, we will give an example to show that the surjectivity assumption cannot be omitted. Using this, we solve the non-surjective isometric extension problem in the negative.     

Surfaces with parallel normalized mean curvature vector

A surfaceM in a Riemannian manifold is said to have parallel normalized mean curvature vector if the mean curvature vector is nonzero and the unit vector in the direction of the mean curvature vector is parallel in the normal bundle. In this paper, it is proved that every analytic surface in a euclideanm-spaceE ^m with parallel normalized mean curvature vector must either lies in aE ⁴ or lies in a hypersphere ofE ^m as a minimal surface. Moreover, it is proved that if a Riemann sphere inE ^m has parallel normalized mean curvature vector, then it lies either in aE ³ or in a hypersphere ofE ^m as a minimal surfaces. Applications to the classification of surfaces with constant Gauss curvature and with parallel normalized mean curvature vector are also given.     

Polynomials in operator space theory

The aim of this article is to start a metric theory of homogeneous polynomials in the category of operator spaces. For this purpose we take advantage of the basic fact that the space P^m(E)$P^m(E)$ of all m-homogeneous polynomials on a vector space E can be identified with the algebraic dual of the m  -th symmetric tensor product ⊗^m,sE${\otimes }^{m,s}E$. Given an operator space V, we study several different types of completely bounded polynomials on V   which form the operator space duals of ⊗^m,sV${\otimes }^{m,s}V$ endowed with related operator structures. Of special interest are what we call Haagerup, Kronecker, and Schur polynomials – polynomials associated with different types of matrix products.     

Deformation of a submanifold in a Euclidean space with fixed gauss image: II

Deformation of an m-dimensional submanifold immersed in a Euclidean n-space is studied when the Gauss image does not move pointwise. The present paper is the third one and contains deformations where the tensor of deformation satisfies some differential equations, normal deformations and deformations of totally-real submanifolds in a complex space.     

Non-Existence of Stable Currents in Hypersurfaces

Let M^m be a compact hypersurface in the Euclidean space E^m+1. In this paper, we study the non-existence of stable integral currents in M^m and its immersed submanifolds. Some vanishing theorems concerning the homology groups of these manifolds are established.AMS Subject Classification (1991): 49Q15 53C40 53C20     

Note on function spaces with the topology of pointwise convergence

The note contains two examples of function spaces C _p (X) endowed with the pointwise topology. The first example is C _p (M), M being a planar continuum, such that C _p (M) ^m is uniformly homeomorphic to C _p (M) ⁿ if and only if m = n. This strengthens earlier results concerning linear homeomorphisms. The second example is a non-Lindelöf function space C _p (X), where X is a monolithic perfectly normal compact space all linearly orderable closed subspaces of which are metrizable. This example is obtained under the additional set-theoretical axiom . This solves a problem of Arhangelskiĭ.     

Visibility queries in a polygonal region

In this paper, we consider the problem of computing the region visible to a query point located in a given polygonal domain. The polygonal domain is specified by a simple polygon with m holes and a total of n vertices. We provide two bounds on the complexity of this problem. One approach constructs a data structure with space complexity O(n²) in time O(n²lgn) and yields a query time of O((1+min(m,|V(q)|))lg²n+m+|V(q)|). Here, V(q) represents the set of vertices of the visibility polygon of a query point q, and |E| denotes the number of edges in the visibility graph. The other approach provides a data structure with space complexity O(min(|E|,mn)+n) in time O(T+|E|+nlgn) with the query time of O(|V(q)|lgn+m). Here, T is the time to triangulate the given polygonal region (which is O(n+mlg¹⁺m) for a small positive constant >0). In both of these approaches, O(m) additive factor in the query time is eliminated with an additional O((min(|E|,mn))²) space and an additional O(m(min(|E|,mn))²) preprocessing time.