論ПОНТРЯГИН示性類,Ⅲ.

<正> 本文繼續以前二文研究微分流形上示性類的拓撲不變性. 本文應用了在[3]一文中首次倡用的方法,完全決定了格拉斯曼流形R_n,m中的平方。由此可知,在一個可微分閉流形上,示性類在法4約化後乃是這個閉流形的拓撲不變量。     

論Понтрягин示性類Ⅴ

<正> 本文是這系列著作中Ⅱ的一個補充.在Ⅱ中(參閱Ⅱ的更正)我們證明了可微分閉流形的某些示性類特別是法3示性類的拓撲不變性.它的證明是隱合的(implicit).本文目的在進一步求得這些示性類用流形同調構造來表示的顧谿(explicit)公式,使我們能就任意可定向的可微分閉流形的這些示性類進行具體的計算.特別可以獲得下述結果:     

Cheeger-gromoll type metrics on the (1,1)-tensor bundles

Using a Riemannian metric on a differentiable manifold, a Cheeger-Gromoll type metric is introduced on the (1,1)-tensor bundle of the manifold. Then the Levi-Civita connection, Riemannian curvature tensor, Ricci tensor, scalar curvature and sectional curvature of this metric are calculated. Also, a para-Nordenian structure on the the (1,1)-tensor bundle with this metric is constructed and the geometric properties of this structure are studied.     

Pseudodistances and Pseudometrics on real and complex manifolds

Summary In this paper we study the relationships between a class of distances and infinitesimal metrics on real and complex manifolds and their behavior under differentiable and holomorphic mappings. Some application to Riemannian and Finsler geometry are given and also new proofs and generalizations of some results of Royden, Harris and Reiffen on Kobayashi and Carathéodory metrics on complex manifolds are obtained. In particular we prove that on every complex manifold (finite or infinite- dimensional) the Kobayashi distance is the integrated form of the corresponding infinitesimal metric.     

Three classes of Weitzenböck manifolds

A Weitzenböck manifold is a triplet defined by a differentiable manifold with a metric g of certain signature and a linear connection with zero curvature tensor and nonzero torsion tensor which is a metric connection with respect to g. The theory of such manifolds is called the “new theory of gravity”. We study properties of three classes of Weitzenböck manifolds and prove some vanishing thorems.     

Twistor spaces of hypercomplex manifolds are balanced

A hypercomplex structure on a differentiable manifold consists of three integrable almost complex structures that satisfy quaternionic relations. If, in addition, there exists a metric on the manifold which is Hermitian with respect to the three structures, and such that the corresponding Hermitian forms are closed, the manifold is said to be hyperkähler. In the paper “Non-Hermitian Yang–Mills connections” [13], Kaledin and Verbitsky proved that the twistor space of a hyperkähler manifold admits a balanced metric; these were first studied in the article “On the existence of special metrics in complex geometry” [17] by Michelsohn. In the present article, we review the proof of this result and then generalize it and show that twistor spaces of general compact hypercomplex manifolds are balanced.     

On fat sub-Riemannian symmetric spaces in codimension three

A sub-Riemannian manifold is a differentiable manifold together with a smooth distribution which is equipped with a Riemannian metric. In this paper we attempt to study sub-Riemannian symmetric spaces (i.e., homogeneous sub-Riemannian manifolds admitting an involutive sub-Riemannian isometry at all points which is a central symmetry when restricted to the distribution) where the associated distribution is a codimension three fat distribution. We obtain a restricted classification theorem in dimension seven and we also construct a class of examples of quaternionic type in varying dimension.     

Some Remarks Concerning Hyperholomorphic B-Manifolds

The authors consider a differentiable manifold with H-structure which is an isomorphic representation of an associative, commutative and unitial algebra. For Riemannian metric tensor fields, the φ-operators associated with r-regular H-structure are introduced. With the help of φ-operators, the hyperholomorphity condition of B-manifolds is established.     

Curvature identities derived from the integral formula for the first Pontrjagin number

We give an integral formula for the first Pontrjagin number of a compact almost Hermitian surface and derive curvature identities from the integral formula based on the fundamental fact that the first Pontrjagin number in the deRham cohomology group is a topological invariant. Further, we provide some applications of the identities.     

Riemannsche Arealstrukturen und multiquadratische Formen

It is well known (and rather trivial to prove) that the square F² of a Finsler norm F:TM on a differentiable manifold M is differentiable at the zero section if and only if F is the norm function of a Riemannian metric. However, the corresponding question for a general p-areal on M is far less trivial to settle and leads to interesting algebraic and combinatorial problems concerning multiquadratic forms. For p=2, the results are closely related to known properties of the curvature tensor of a Riemannian metric.     

論Понтрягин示性類(Ⅳ)

<正> 前言 在前一文中,我們曾應用Steenrod冪以證明一個可微分閉流形上法示性類的拓撲不變性.本文的目的則在提供一個不同的方法,不假助於Steenrod冪以證明同一事實.同樣的方法,亦可用於Stiefel-Whitney示性類,而由此獲得這些類的拓撲不變性的一個不同證明,而在證明中避免用到Steenrod     

A characterization of the Â-genus as a linear combination of Pontrjagin numbers

We show in this short note that if a rational linear combination of Pontrjagin numbers vanishes on all simply-connected 4k-dimensional closed connected and oriented spin manifolds admitting a Riemannian metric whose Ricci curvature is nonnegative and not identically zero, then this linear combination must be a multiple of the Â-genus, which improves a result of Gromov and Lawson. Our proof combines an idea of Atiyah and Hirzebruch and the celebrated Calabi–Yau theorem.     

Rozansky–Witten invariants via Atiyah classes

Recently, L. Rozansky and E. Witten associated to any hyper-Kähler manifold X a system of weights (numbers, one for each trivalent graph) and used them to construct invariants of topological 3-manifolds. We give a simple cohomological definition of these weights in terms of the Atiyah class of X (the obstruction to the existence of a holomorphic connection). We show that the analogy between the tensor of curvature of a hyper-Kähler metric and the tensor of structure constants of a Lie algebra observed by Rozansky and Witten, holds in fact for any complex manifold, if we work at the level of cohomology and for any Kähler manifold, if we work at the level of Dolbeault cochains. As an outcome of our considerations, we give a formula for Rozansky–Witten classes using any Kähler metric on a holomorphic symplectic manifold.     

On the existence of almost contact structure and the contact magnetic field

We give a simple proof of the existence of an almost contact metric structure on any orientable 3-dimensional Riemannian manifold (M ³, g) with the prescribed metric g as the adapted metric of the almost contact metric structure. By using the key formula for the structure tensor obtained in the proof this theorem, we give an application which allows us to completely determine the magnetic flow of the contact magnetic field in any 3-dimensional Sasakian manifold.     

Anti-Kählerian manifolds

An anti-Kählerian manifold is a complex manifold with an anti-Hermitian metric and a parallel almost complex structure. It is shown that a metric on such a manifold must be the real part of a holomorphic metric. It is proved that all odd Chern numbers of an anti-Kählerian manifold vanish and that complex parallelisable manifolds (in particular the factor space G/D of a complex Lie group G over the discrete subgroup D ) are anti-Kählerian manifolds. A method of generating new solutions of Einstein equations by using the theory of anti-Kählerian manifolds is presented.     

Generalized Ricci curvature bounds for three dimensional contact subriemannian manifolds

Measure contraction property is one of the possible generalizations of Ricci curvature bound to more general metric measure spaces. In this paper, we discover necessary and sufficient conditions for a three dimensional contact subriemannian manifold to satisfy this property.     

The hypoelliptic Laplacian,analytic torsion and Cheeger–Müller Theorem

The purpose of this Note is to prove a formula relating the hypoelliptic Ray–Singer metric and the Milnor metric on the determinant of the cohomology of a compact Riemannian manifold by a Witten-like deformation of the hypoelliptic Laplacian in de Rham theory.     

An Intrinsic Geometric Formulation of the Equilibrium Equations in Continuum Mechanics

The theory of invariant continuum mechanics is based on the concept that forces and stresses are defined as elements of the cotangent bundle of the configuration manifold. While body and physical space are modeled as differentiable manifolds, the infinite dimensional configuration manifold is given by all configurations of the body in the physical space. In this paper a virtual work principle is postulated which leads together with an induced traction stress and Stokes' theorem directly to the local equilibrium equations and the traction boundary conditions. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)