复射影空间中具有平行平均曲率向量的全实叶的Pinching定理

本文研究了复射影空间中具有平行平均曲率向量的黎曼叶状结构(F).利用Nalcagawa 和Takagi的计算散度的方法,得到了复射影窄问中具有平行平均曲率向量的黎曼叶状结构(F)上向量场的散度,证明了其上的一个整体Pinching定理,从而将复射影空间中任何具有极小法平面场的调和叶的仵质推广到复射影窄间中具有平行平均曲率向量的黎曼叶状结构(F)上.     

关于光滑函数类上的最优取样

§1.问题的提出 设X_1,X_2,X_3是实或复的线性赋范空间,KX_1为零点对称凸集,S是X_1→X_2的线性算子.I是X_1→X_3的线性算子。任取x∈K,Ix称为x的信息,I是K的一信息算子。根据[1],S在K上利用信息I的最优回复问题中一个基本的量是确定信息直径     

体上线性映射的子空间的维数及其应用

本文给出体上左向量空间的线性映射的某些子空间的维数恒等式,并讨论了它在体上矩阵秩的理论上的应用,其中一个有趣的应用是,由体上矩阵秩的恒等式来刻划体上某些矩阵的特征性质。 以下设Ω是一个体。对Ω上左向量空间V映入Ω上左向量空间V′的线性映射σ:V V~σ,记σ的核空间为:     

反对称张量空间中可合元素的一些性质

<正> 本文讨论反对称张量空间可合元素的必要条件,用可合元素的 Plücker坐标表示相应子空间的方法以及子空间正交与可合元素之间的关系.设 V 是一个特征为 0 的域 R 上的 n 维向量空间.对于1≤m≤n,(?)V 是 m 阶     

一个辛不动点定理

本文利用临界点理论,讨论了环面与复射影空间乘积 T~(2n)×CP~m 上正合辛同胚的不动点个数的下界估计问题,部分地回答了 V.I.Arnold 关于正合辛同胚不动点数问题的猜测.     

强向量拟均衡问题的存在性

设X是一个实的Hausdorff拓扑向量空间,Y是一个实的局部凸向量空间,C是Y中的闭凸锥,K X是一个紧子集.FX×X→Y是一个双向量函数,GK→2K是一个集合值映射.我们考虑下面的强拟均衡问题存在x∈G(x),使得对任意的y∈G(x),成立F(x,y)∈C.本文证明了当F是半连续时,上述问题解的存在性结论.     

关于非齐次线性方程组解的结构的进一步讨论

一、子空间的陪集定义1.设V是数域F上的向量空间,W是V的子空间。若v是V中任意向量,把和v w(w∈W)组成的集记作v W,即v W={v w|w∈W},则这些集称为V中W的陪集。 容易证明下面定理 定理1.V中W的陪集将V分成互不相交的集,即:(ⅰ)任何两个陪集u W与v W或重合或不相交;(ⅱ)每个v∈V属于一个陪集,事实上v∈v W     

最佳同时逼近的特征

<正> 设X是赋范线性空间(实或复),V、F是X的子集,F有界,如果ν_o∈V满足则称ν_o是V对F的最佳同时逼近,其全体记为P_v(F). 最佳同时逼近近年来已有不少文献进行研究,J.H.Freilich,H.w.Mclaughlin     

关于二次型零点的注记

给出复系数和实系数n元二次型零点向量组的秩,以及最大零点子空间的维数.证明了正负惯性指数为p,q的实二次型的最大零点子空间的维数为n-max{p,q},以及秩为r的复二次型的最大零点子空间的维数为■.     

示性式的超渡(续完)

<正> 7.Grassmann流形与陈类 设为复N维向量酉空间中全体n维子空间所成的Grassmann流形,它是一个齐性空间     

On holomorphic immersions into K(a)hler manifolds of constant holomorphic sectional curvature

We study holomorphic immersions f: X → M from a complex manifold X into a Kahler manifold of constant holomorphic sectional curvature M, i.e. a complex hyperbolic space form, a complex Euclidean space form, or the complex projective space equipped with the Fubini-Study metric. For X compact we show that the tangent sequence splits holomorphically if and only if f is a totally geodesic immersion. For X not necessarily compact we relate an intrinsic cohomological invariant p(X) on X, viz. the invariant defined by Gunning measuring the obstruction to the existence of holomorphic projective connections, to an extrinsic cohomological invariant v(f)measuring the obstruction to the holomorphic splitting of the tangent sequence. The two invariants p(X) and v(f) are related by a linear map on cohomology groups induced by the second fundamental form.In some cases, especially when X is a complex surface and M is of complex dimension 4, under the assumption that X admits a holomorphic projective connection we obtain a sufficient condition for the holomorphic splitting of the tangent sequence in terms of the second fundamental form.     

An integral invariant from the view point of locally conformally Kähler geometry

In this article we study an integral invariant which obstructs the existence on a compact complex manifold of a volume form with the determinant of its Ricci form proportional to itself, in particular obstructs the existence of a Kähler-Einstein metric, and has been studied since 1980s. We study this invariant from the view point of locally conformally Kähler geometry. We first see that we can define an integral invariant for coverings of compact complex manifolds with automorphic volume forms. This situation typically occurs for locally conformally Kähler manifolds. Secondly, we see that this invariant coincides with the former one. We also show that the invariant vanishes for any compact Vaisman manifold.     

On Compact Solvability of Differentials of an Elliptic Differential Complex

We find sufficient conditions for compact solvability of differentials of an elliptic differential complex on a noncompact Riemannian manifold. As the main example we consider the de Rham complex of differential forms on a manifold with cylindrical ends.     

On vanishing theorems for Higgs bundles

We introduce the notion of Hermitian Higgs bundle as a natural generalization of the notion of Hermitian vector bundle and we study some vanishing theorems concerning Hermitian Higgs bundles when the base manifold is a compact complex manifold. We show that a first vanishing result, proved for these objects when the base manifold was Kähler, also holds when the manifold is compact complex. From this fact and some basic properties of Hermitian Higgs bundles, we conclude several results. In particular we show that, in analogy to the classical case, there are vanishing theorems for invariant sections of tensor products of Higgs bundles. Then, we prove that a Higgs bundle admits no nonzero invariant sections if there is a condition of negativity on the greatest eigenvalue of the Hitchin–Simpson mean curvature. Finally, we prove that the invariant sections of certain tensor products of a weak Hermitian–Yang–Mills Higgs bundle are all parallel in the classical sense.     

Una applicazione delle correnti invarianti ai gruppi di Lie compatti

Riassunto Nella presente nota si introduce il complesso delle correnti invarianti su un gruppo di Lie compatto connesso e si prova che è omologicamente equivalente al complesso delle catene di Koszul e a quello delle forme invarianti, Ciò permette di riformulare in linguaggio unitario risultati di Chevalley e Eilenberg e di Koszul sulle proprietà omologiche dei gruppi di Lie.

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Summary In this note we introduce the complex of the invariant currents on a connected compact Lie group. We prove that it is homologically equivalent with the complex of the Koszul chains and with the complex of the invariant forms, so that we can restate in a unitary manner some results of Chevalley-Eilenberg and of Loszul on homological properties of Lie groups.

     

Stability of complex hyperbolic space under curvature-normalized Ricci flow

Using the maximal regularity theory for quasilinear parabolic systems, we prove two stability results of complex hyperbolic space under the curvature-normalized Ricci flow in complex dimensions two and higher. The first result is on a closed manifold. The second result is on a complete noncompact manifold. To prove both results, we fully analyze the structure of the Lichnerowicz Laplacian on complex hyperbolic space. To prove the second result, we also define suitably weighted little Hölder spaces on a complete noncompact manifold and establish their interpolation properties.     

Hermitian structures on six-dimensional nilmanifolds

Let (J,g) be a Hermitian structure on a six-dimensional compact nilmanifold M with invariant complex structure J and compatible metric g, which is not required to be invariant. We show that, up to equivalence of the complex structure, the strong Kahler with torsion structures (J,g) on M are parametrized by the points in a subset of the Euclidean space, in particular, the region inside a certain ovaloid corresponds to such structures on the Iwasawa manifold and the region outside to strong Kahler with torsion structures with nonabelian J on the nilmanifold where H³ is the Heisenberg group. A classification of six-dimensional nilmanifolds admitting balanced Hermitian structures (J,g) is given, and as an application we classify the nilmanifolds having invariant complex structures which do not admit Hermitian structure with restricted holonomy of the Bismut connection contained in SU(3). It is also shown that on the nilmanifold the balanced condition is not stable under small deformations. Finally, we prove that a compact quotient of where H(2,1) is the five-dimensional generalized Heisenberg group, is the only six-dimensional nilmanifold having locally conformal Kahler metrics, and the complex structures underlying such metrics are all equivalent. Moreover, this nilmanifold is a Vaisman manifold for any invariant locally conformal Kahler metric.     

A manifold that does not contain a compact core

A core of a (noncompact) manifold is a submanifold with the property that the inclusion of the submanifold into the manifold is a homotopy equivalence. It is shown by example that a manifold may fail to contain a compact core even though the manifold has the homotopy type of a finite complex.