S~5上仿Blaschke张量的特征值为常数的超曲面

设x:M→S~(n+1)是(n+1)-维单位球面上不含脐点的超曲面,在S~(n+1)的Moebius变换群下浸入x的四个基本不变量是:一个黎曼度量g称为Moebius度量;一个1-形式Φ称为Moebius形式;一个对称的(0,2)张量A称为Blaschke张量和一个对称的(0,2)张量B称为Moebius第二基本形式.对称的(0,2)张量D=A+λB也是Moebius不变量,其中λ是常数,D称为浸入x的仿Blaschke张量.李海中和王长平研究了满足条件:(i)Φ=0;(ii)A+λB+μg=0的超曲面,其中λ和μ都是函数,他们证明了λ和μ都是常数,并且给出了这类超曲面的分类,也就是在Φ=0的条件下D只有一个互异的特征值的超曲面的分类.本文对S~5上满足如下条件的超曲面进行了完全分类:(i)Φ=0,(ii)对某常数λ,D具有常数特征值.     

单位球面上仿Blaschke张量的特征值为常数的超曲面

设x:M~n→S~(n+1)是(n+1)-维单位球面上不含脐点的超曲面,在S~(n+1)的Moebius变换群下浸入x的四个基本不变量是:一个黎曼度量g称为Moebius度量;一个1-形式Φ称为Moebius形式;一个对称的(0,2)张量A称为Blaschke张量和一个对称的(0,2)张量B称为Moebius第二基本形式.对称的(0,2)张量D=A+λB也是Moebius不变量,称为浸入x的仿Blaschke张量,其中λ是常数,仿Blaschke张量的特征值称为仿Blaschke特征值.李海中和王长平(2003)研究了满足如下条件的超曲面:(i)Φ=0;(ii)存在可微函数λ和μ,使A+λg+μB=0.他们证明了λ和μ都是常数,并且给出了这类超曲面的分类,也就是D的特征值全相等的超曲面的分类.本文对满足如下条件的超曲面进行了分类:(i)Φ=0,(ii)对某一个常数λ,D具有两个互异的常数特征值.     

A characterization of Moebius isoparametric hypersurfaces of the sphere

We consider a proper, umbilic-free immersion of an n-dimensional manifold M in the sphere S ⁿ⁺¹. We show that M is a Moebius isoparametric hypersurface if, and only if, it is a cyclide of Dupin or a Dupin hypersurface with constant Moebius curvature.     

Submanifolds with constant Möbius scalar curvature in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

 Let M ^m be a m-dimensional submanifold in the n-dimensional unit sphere S ⁿ without umbilic point. Two basic invariants of M ^m under the M?bius transformation group of S ⁿ are a 1-form Φ called M?bius form and a symmetric (0,2) tensor A called Blaschke tensor. In this paper, we prove the following rigidity theorem: Let M ^m be a m-dimensional (m≥3) submanifold with vanishing M?bius form and with constant M?bius scalar curvature R in S ⁿ , denote the trace-free Blaschke tensor by . If , then either ||?||≡0 and M ^m is M?bius equivalent to a minimal submanifold with constant scalar curvature in S ⁿ ; or and M ^m is M?bius equivalent to in for some c≥0 and . Received: 15 May 2002 / Revised version: 3 February 2003 Published online: 19 May 2003 RID="*" ID="*" Partially supported by grants of CSC, NSFC and Outstanding Youth Foundation of Henan, China. RID="†" ID="†" Partially supported by the Alexander Humboldt von Stiftung and Zhongdian grant of NSFC. Mathematics Subject Classification (2000): Primary 53A30; Secondary 53B25     

Locally symmetric complex affine hypersurfaces

In this paper we study non-degenerate locally symmetric complex affine hypersurfaces Mⁿ of the complex affine space, i.e. hypersurfaces satisfying R=0, where is the affine connection induced on Mⁿ by the complex affine structure on the complex affine space, and R is the curvature tensor of . We classify the non-degenerate locally symmetric hypersurfaces Mⁿ, n > 2, and the minimal non-degenerate locally symmetric hypersurfaces Mⁿ, n > 1.Aspirant N.F.W.O. (Belgium)     

Classification of hypersurfaces with parallel Möbius second fundamental form in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis>+1</Superscript>

Let M ⁿ(n ≥ 2) be an immersed umbilic-free hypersurface in the (n+1)-dimensional unit sphere S ⁿ⁺¹. Then M ⁿ is associated with a so-called Möbius metric g, and a Möbius second fundamental form B which are invariants of M ⁿunder the Möbius transformation group of S ⁿ⁺¹. In this paper, we classify all umbilic-free hypersurfaces with parallel Möbius second fundamental form.     

A characterization of Moebius isoparametric hypersurfaces of the sphere

We consider a proper, umbilic-free immersion of an n-dimensional manifold M in the sphere S ⁿ⁺¹. We show that M is a Moebius isoparametric hypersurface if, and only if, it is a cyclide of Dupin or a Dupin hypersurface with constant Moebius curvature.     

Maximal number of distinct H-eigenpairs for a two-dimensional real tensor

Based on the generalized characteristic polynomial introduced by J. Canny in Generalized characteristic polynomials [J. Symbolic Comput., 1990, 9(3): 241–250], it is immediate that for any m-order n-dimensional real tensor, the number of distinct H-eigenvalues is less than or equal to n(m?1) ^n?1. However, there is no known bounds on the maximal number of distinct Heigenvectors in general. We prove that for any m ? 2, an m-order 2-dimensional tensor A exists such that A has 2(m ? 1) distinct H-eigenpairs. We give examples of 4-order 2-dimensional tensors with six distinct H-eigenvalues as well as six distinct H-eigenvectors. We demonstrate the structure of eigenpairs for a higher order tensor is far more complicated than that of a matrix. Furthermore, we introduce a new class of weakly symmetric tensors, called p-symmetric tensors, and show under certain conditions, p-symmetry will effectively reduce the maximal number of distinct H-eigenvectors for a given two-dimensional tensor. Lastly, we provide a complete classification of the H-eigenvectors of a given 4-order 2-dimensional nonnegative p-symmetric tensor. Additionally, we give sufficient conditions which prevent a given 4-order 2-dimensional nonnegative irreducible weakly symmetric tensor from possessing six pairwise distinct H-eigenvectors.     

Isotropic affine hypersurfaces of dimension 5

We study affine hypersurfaces M   which have isotropic difference tensor. Note that any surface always has isotropic difference tensor. Therefore, we may assume that n>2$n>2$. Such hypersurfaces have previously been studied by the first author and M. Djoric in [1] under the additional assumption that M is an affine hypersphere. Here we study the general case. As for affine spheres, we first show that isotropic affine hypersurfaces which are not congruent to quadrics are necessarily 5, 8, 14 or 26 dimensional. From this, we also obtain a complete classification in dimension 5.     

Immersed Hypersurfaces in the Unit Sphere S^m＋1 with Constant Blaschke Eigenvalues

For an immersed submanifold x ： M^m→ Sn in the unit sphere S^n without umbilics, an eigenvalue of the Blaschke tensor of x is called a Blaschke eigenvalue of x. It is interesting to determine all hypersurfaces in Sn with constant Blaschke eigenvalues. In this paper, we are able to classify all immersed hypersurfaces in S^m＋1 with vanishing MSbius form and constant Blaschke eigenvalues, in case （1） x has exact two distinct Blaschke eigenvalues, or （2） m = 3. With these classifications, some interesting examples are also presented.     

Drazin inverse of partitioned matrices in terms of Banachiewicz-Schur forms

Let be a partitioned matrix, where A and D are square matrices. Denote the Drazin inverse of A by A^D. The purpose of this paper is twofold. Firstly, we develop conditions under which the Drazin inverse of M having generalized Schur complement, S=D-CA^DB, group invertible, can be expressed in terms of a matrix in the Banachiewicz-Schur form and its powers. Secondly, we deal with partitioned matrices satisfying rank(M)=rank(A^D)+rank(S^D), and give conditions under which the group inverse of M exists and a formula for its computation.     

Regular space-like hypersurfaces in S<Stack><Subscript>1</Subscript><Superscript><Emphasis Type="Italic">m</Emphasis>+1</Superscript></Stack> with parallel para-Blaschke tensors

In this paper, we give a complete conformal classification of the regular space-like hypersurfaces in the de Sitter Space S ^m+1 ₁ with parallel para-Blaschke tensors.     

Embedding of a pseudo-residual design into a Möbius plane

Let $\text{U}$ be a class of subsets of a finite set X. Elements of $\text{U}$ are called blocks. Let v, t and λ₁, 0 ? i ? t, be nonnegative integers, and K be a subset of nonnegative integers such that every member of K is at most v. A pair (X, U) is called a (λ₀, λ₁,…, λ_t; K, υ)t-design if (1) |X| = υ, (2) every i-subset of X is contained in exactly λ_t blocks, 0 ? i ? t, and (3) for every block A in $\text{U}$, |A| ?K. It is well-known that if K consists of a singleton k, then λ₀,…, λ_{t ? 1} can be determined from υ, t, k and λ_t. Hence, we shall denote a (λ₀,…, λ_t; {k}, υ)t-design by S_λ(t, k, υ), where λ = λ_t. A Möbius plane M is an S₁(3, q + 1, q² + 1), where q is a positive integer. Let A be a fixed block in M. If A is deleted from M together with the points contained in A, then we obtain a residual design M′ with parameters λ₀ = q³ + q ? 1, λ₁ = q² + q, λ₂ = q + 1, λ₃ = 1, K = {q + 1, q, q ? 1}, and υ = q² ? 1. We define a design to be a pseudo-block-residual design of order q (abbreviated by PBRD(q)) if it has these parameters. We consider the reconstruction problem of a Möbius plane from a given PBRD(q). Let B and B′ be two blocks in a residual design M′. If B and B′ are tangent to each other at a point x, and there exists a block C of size q + 1 such that C is tangent to B at x and is secant to B′, then we say B is r-tangent to B′ at x. A PBRD(q) is said to satisfy the r-tangency condition if for every block B of size q, and any two points x and y not in B, there exists at most one block which is r-tangent to B and contains x and y. We show that any PBRD(q)D can be uniquely embedded into a Möbius plane if and only if D satisfies the r-tangency condition.     

Kronecker multiplicities in the (k, ℓ) hook are polynomially bounded

The problem of decomposing the Kronecker product of S _n characters is one of the last major open problems in the ordinary representation theory of the symmetric group S _n . In this note λ and µ are partitions of n, n goes to infinity, and we prove upper and lower polynomial bounds for the multiplicities of the Kronecker product χ^λ ? χ^µ, where for some fixed k and ? both partitions λ and µ are in the (k, ?) hook.     

k-Potence preserving maps without the linearity and surjectivity assumptions

Let M_n be the space of all n × n complex matrices, and let Γ_n be the subset of M_n consisting of all n × n k-potent matrices. We denote by Ψ_n the set of all maps on M_n satisfying A − λB ∈ Γ_n if and only if ?(A) − λ?(B) ∈ Γ_n for every A,B ∈ M_n and λ ∈ C. It was shown that ? ∈ Ψ_n if and only if there exist an invertible matrix P ∈ M_n and c ∈ C with c^k−1 = 1 such that either ?(A) = cPAP⁻¹ for every A ∈ M_n, or ?(A) = cPA^TP⁻¹ for every A ∈ M_n.     

A note on k-potence preserving maps

Let M_n be the algebra of all n×n complex matrices and Γ_n the set of all k-potent matrices in M_n. Suppose ?:M_n→M_n is a map satisfying A-λB∈Γ_n implies ?(A)-λ?(B)∈Γ_n, where A, B∈M_n, λ∈C. Then either ? is of the form ?(A)=cTAT^-1, A∈M_n, or ? is of the form ?(A)=cTA^tT^-1, A∈M_n, where T∈M_n is an invertible matrix, c∈C satisfies c^k=c.     

A conjecture involving generalized matrix functions

This paper resolves the following conjecture of R. Merris: Let d^G_λ be the generalized matrix function determined by a subgroup G of the symmetric group S_m and an irreducible complex character λ of G. If A, B, and A?B are m-square positive semidefinite hermitian m-square matrices and d^G_λ(A)=d^G_λ(B)≠0, then A=B.     

Willmore hypersurfaces with constant Möbius curvature in R n+1

For an immersed hypersurface ${f : M^n \rightarrow R^{n+1}}$ without umbilical points, one can define the Möbius metric g on f which is invariant under the Möbius transformation group. The volume functional of g is a generalization of the well-known Willmore functional, whose critical points are called Willmore hypersurfaces. In this paper, we prove that if a n-dimensional Willmore hypersurfaces ${(n \geq 3)}$ has constant sectional curvature c with respect to g, then c = 0, n = 3, and this Willmore hypersurface is Möbius equivalent to the cone over the Clifford torus in ${S^{3} \subset R^{4}}$ . Moreover, we extend our previous classification of hypersurfaces with constant Möbius curvature of dimension ${n \ge 4}$ to n = 3, showing that they are cones over the homogeneous torus ${S^1(r) \times S^1(\sqrt{1 - r^2}) \subset S^3}$ , or cylinders, cones, rotational hypersurfaces over certain spirals in the space form R ², S ², H ², respectively.     

Simultaneous block diagonalization of two real symmetric matrices

The simultaneous diagonalization of two real symmetric (r.s.) matrices has long been of interest. This subject is generalized here to the following problem (this question was raised by Dr. Olga Taussky-Todd, my thesis advisor at the California Institute of Technology): What is the first simultaneous block diagonal structure of a nonsingular pair of r.s. matrices ? For example, given a nonsingular pair of r.s. matrices S and T, which simultaneous block diagonalizations X′SX = diag(A₁, , A_k), X′TX = diag(B₁,, B_k) with dim A_i = dim B_i and X nonsingular are possible for 1 ? k ? n; and how well defined is a simultaneous block diagonalization for which k, the number of blocks, is maximal? Here a pair of r.s. matrices S and T is called nonsingular if S is nonsingular.If the number of blocks k is maximal, then one can speak of the finest simultaneous block diagonalization of S and T, since then the sizes of the blocks A_i are uniquely determined (up to permutations) by any set of generators of the pencil P(S, T) = {aS + bT|a, tb ε R} via the real Jordan normal form of S^?1T. The proof uses the canonical pair form theorem for nonsingular pairs of r.s. matrices. The maximal number k and the block sizes dim A_i are also determined by the factorization over C of ? (λ, μ) = det(λS + μT) for λ, μ ε R.