Realizing characteristic classes by manifolds

Let BG be the classifying space for stable spherical fibrations, and let V be a finite dimensional vector subspace of the cohomology algebra H¹(BG; $\text{Z}$₂). We prove that V may be realized by a Poincaré duality space P, which means that if v:P→BG is the Spivak fibration, then v¹ maps V isomorphically onto its image in H¹(P;$\text{Z}$₂). By construction, P is the product of a certain Grassman manifold and a spherical fiber space over a closed smooth manifold.     

A note on characteristic classes

This paper studies the relationship between the sections and the Chern or Pontrjagin classes of a vector bundle by the theory of connection. Our results are natural generalizations of the Gauss-Bonnet Theorem.     

Constancy of p-harmonic maps of finite q-energy into non-positively curved manifolds

We investigate p-harmonic maps, p ≥ 2, from a complete non-compact manifold into a non-positively curved target. First, we establish a uniqueness result for the p-harmonic representative in the homotopy class of a constant map. Next, we derive a Caccioppoli inequality for the energy density of a p-harmonic map and we prove a companion Liouville type theorem, provided the domain manifold supports a Sobolev–Poincaré inequality. Finally, we obtain energy estimates for a p-harmonic map converging, with a certain speed, to a given point.      

A note on the characteristic rank of null-cobordant manifolds

We give an upper bound for the characteristic rank of smooth closed null-cobordant manifolds and apply it to the Grassmann manifolds of oriented vector subspaces in Euclidean space. The bound turns out to be sharp: it coincides with the exact value of characteristic rank for the Grassmann manifolds of oriented 2-dimensional vector subspaces in odd-dimensional Euclidean space.     

Lax定理与连续左逆的存在性

本文在局部凸空间的框架下,研究了一般线性算子（未必连续）存在连续左逆的条件,从而获得了Lax定理的一系列新的推广.     

A note on the cohomological below -pinching theorem

By making use of a theorem of Toda, we establish a sharper version of the below -pinching theorem of Abresch and Meyer.

     

A note on the conjugacy classes of non-cyclic subgroups

Let G be a finite group and δ(G) denote the number of conjugacy classes of all non-cyclic subgroups of G. The symbol π(G) denotes the set of the prime divisors of |G|. In [7 Meng, W., Li, S. R. (2014). Finite groups with few conjugacy classes of non-cyclic subgroups. Scientia Sin. Math. 44:939–944.[Crossref] , [Google Scholar]], Meng and Li showed the inequality δ(G)≥2^|π(G)|?2, where G is non-cyclic solvable group. In this paper, we describe the finite groups G such that δ(G) = 2^|π(G)|?2. Another aim of this paper would show δ(G)≥M(G)+2 for unsolvable groups G and the equality holds ?G?A₅ or SL(2,5), where M(G) denotes the number of conjugacy classes of all maximal subgroups of G.     

Equivalence of domains arising from duality of orbits on flag manifolds

S. Gindikin and the author defined a - invariant subset of for each -orbit on every flag manifold and conjectured that the connected component of the identity would be equal to the Akhiezer-Gindikin domain if is of non-holomorphic type by computing many examples. In this paper, we first prove this conjecture for the open -orbit on an ``arbitrary' flag manifold generalizing the result of Barchini. This conjecture for closed was solved by J. A. Wolf and R. Zierau for Hermitian cases and by G. Fels and A. Huckleberry for non-Hermitian cases. We also deduce an alternative proof of this result for non-Hermitian cases.

     

Equivalence of domains arising from duality of orbits on flag manifolds II

S. Gindikin and the author defined a - invariant subset of for each -orbit on every flag manifold and conjectured that the connected component of the identity would be equal to the Akhiezer-Gindikin domain if is of nonholomorphic type. This conjecture was proved for closed in the works of J. A. Wolf, R. Zierau, G. Fels, A. Huckleberry and the author. It was also proved for open by the author. In this paper, we prove the conjecture for all the other orbits when is of non-Hermitian type.

     

Equivalence of domains arising from duality of orbits on flag manifolds III

In Gindikin and Matsuki 2003, we defined a - invariant subset of for each -orbit on every flag manifold and conjectured that the connected component of the identity would be equal to the Akhiezer-Gindikin domain if is of nonholomorphic type. This conjecture was proved for closed in Wolf and Zierau 2000 and 2003, Fels and Huckleberry 2005, and Matsuki 2006 and for open in Matsuki 2006. It was proved for the other orbits in Matsuki 2006, when is of non-Hermitian type. In this paper, we prove the conjecture for an arbitrary non-closed -orbit when is of Hermitian type. Thus the conjecture is completely solved affirmatively.

     

A note on the duality mapping of a locally uniformly convex Banach space

Let X be a real locally uniformly convex Banach space with normalized duality mapping J:X→2^X*. The purpose of this note is to show that for every R>0 and every x₀X there exists a function , which is nondecreasing and such that (r)>0 for r>0,(0)=0 and

for all . Simply, it is shown that the necessity part of the proof of the original analogous necessary and sufficient condition of Prüß, for real uniformly convex Banach spaces, goes over equally well in the present setting. This is a natural setting for the study of many existence problems in accretive and monotone operator theories.     

A note on the instability of embeddings of Cauchy-Riemann manifolds

We prove that there are compact strictly pseudoconvex CR manifolds, embedded into some Euclidean space, that admit small deformations that are also embeddable but their embeddings cannot be chosen close to the original embedding. Both authors were partially supported by NSF grants.     

A short note on canonical topologies on the real line

The purpose of this note is to study the relationships between Lowen functor, para-Lowen functor and Rodabaugh's dual L-topology on the real line. The conclusion which they coincide in the case of a Hutton algebra is obtained.     

A note on fixed point sets in CAT(0) spaces

We show that the fixed point set of a quasi-nonexpansive selfmap of a nonempty convex subset of a CAT(0) space is always closed, convex and contractible. Moreover, we give a construction of a continuous selfmap of a CAT(0) space whose fixed point set is prescribed.     

On the dual Stiefel-Whitney classes of some Grassmann manifolds

We present some non-vanishing dual Stiefel-Whitney classes of the Grassmann manifolds O(n)/O(4) × O(n − 4) for n = 2 ^s + 2 and n = 2 ^s + 3 (s ≧ 3), providing a supplement to results of Hiller, Stong, and Oproiu. Some applications are also mentioned. Part of this research was carried out while the first author was a member of three research teams supported in part by the grant agencies VEGA and APVV, and the second author was a member of a research team supported in part by VEGA.     

一类局部凸空间及其映照特征

称局部凸空间（E,(?)0）为WCM空间若对于任何弱于(?)0的局部凸拓扑(?),（E,(?)）与（E,(?)0）具相同的弱紧圆凸集.本文研究了WCM空间的存在性及其与其他类型局部凸空间之间的关系,还给出了WCM空间的一种映照特征.     

On the zeros of functions in Bergman spaces and in some other related classes of functions

A well-known theorem of H.S. Shapiro and A.L. Shields implies that if f?0 is a function which belongs to the Bergman space Ap (0<p<∞) and {zk} is a sequence of zeros of f which is contained in a Stolz angle, then {zk} satisfies the Blaschke condition. In this paper we improve this result. We consider a large class of regions contained in the unit disc D which touch ∂D at a point ξ tangentially and we prove that the mentioned result remains true if we substitute a Stolz angle by any of these regions of tangential approach.     

The Cartan model of the canonical vector bundles over the Grassmannian manifolds

We give a representation of the canonical vector bundles

Open image in new window

over the Grassmannian manifolds G(n, p) as noncompact symmetric affine spaces together with their Cartan model in the group of the Euclidean motions SE(n).