Parallel Spinors and Holonomy Groups on Pseudo-Riemannian Spin Manifolds

We describe the possible holonomy groups of simply connected irreducible non-locally symmetric pseudo-Riemannian spin manifolds which admit parallel spinors.     

Spacelike Graphs with Parallel Mean Curvature in Pseudo-Riemannian Product Manifolds

The author introduces the w-function defined on the considered spacelike graph M.Under the growth conditions w = o(log z) and w = o(r),two Bernstein type theorems for M in Rmn+ mare got,where z and r are the pseudo-Euclidean distance and the distance function on M to some fixed point respectively.As the ambient space is a curved pseudoRiemannian product of two Riemannian manifolds(Σ1,g1) and(Σ2,g2) of dimensions n and m,a Bernstein type result for n = 2 under some curvature conditions on Σ1 and Σ2 and the growth condition w = o(r) is also got.As more general cases,under some curvature conditions on the ambient space and the growth condition w = o(r) or w = o(√r),the author concludes that if M has parallel mean curvature,then M is maximal.     

On Manifolds Whose Tangent Bundle is Big and 1-Ample

A line bundle over a complex projective variety is called bigand 1-ample if a large multiple of it is generated by globalsections and a morphism induced by the evaluation of the spanningsections is generically finite and has at most 1-dimensionalfibers. A vector bundle is called big and 1-ample if the relativehyperplane line bundle over its projectivisation is big and1-ample. The main theorem of the present paper asserts that any complexprojective manifold of dimension 4 or more, whose tangent bundleis big and 1-ample, is equal either to a projective space orto a smooth quadric. Since big and 1-ample bundles are ‘almost’ample, the present result is yet another extension of the celebratedMori paper ‘Projective manifolds with ample tangent bundles’(Ann. of Math. 110 (1979) 593–606). The proof of the theorem applies results about contractionsof complex symplectic manifolds and of manifolds whose tangentbundles are numerically effective. In the appendix we re-provethese results. 2000 Mathematics Subject Classification 14E30,14J40, 14J45, 14J50.     

Hypersurfaces with Parallel Ricci Tensor in Spaces of Constant Curvature

Hypersurfaces with parallel Ricci tensor in spaces of constant curvature are classified. The main tool is a generalization of Moore’s decomposition theorem for isometric immersions.     

Indefinite Kasparov Modules and Pseudo-Riemannian Manifolds

We present a definition of indefinite Kasparov modules, a generalisation of unbounded Kasparov modules modelling non-symmetric and non-elliptic (e.g. hyperbolic) operators. Our main theorem shows that to each indefinite Kasparov module we can associate a pair of (genuine) Kasparov modules, and that this process is reversible. We present three examples of our framework: the Dirac operator on a pseudo-Riemannian spin manifold (i.e. a manifold with an indefinite metric); the harmonic oscillator; and the construction via the Kasparov product of an indefinite spectral triple from a family of spectral triples. This last construction corresponds to a foliation of a globally hyperbolic spacetime by spacelike hypersurfaces.     

Levi-Civita Connections on Degenerate Pseudo-Riemannian Manifolds

Smooth manifolds with degenerate symmetric bilinear form of constant signature on tangent spaces are studied. Such manifolds are used in physical theories of unified interactions including gravitation. Necessary and sufficient conditions for the existence of a Levi-Civita connection on such a manifold are given. Bibliography: 6 titles.     

Computing Asymptotic Invariants with the Ricci Tensor on Asymptotically Flat and Asymptotically Hyperbolic Manifolds

We prove in a simple and coordinate-free way the equivalence between the classical definitions of the mass or of the center of mass of an asymptotically flat manifold and their alternative definitions depending on the Ricci tensor and conformal Killing fields. This enables us to prove an analogous statement in the asymptotically hyperbolic case.     

Ricci曲率平行的一类Riemann流形的C∞紧性

本文证明,在Gromov-Hausdorff拓扑下,Ricci曲率平行,截面曲率和单一半径有下界,体积有上界的Riemann流形的集合是c^∞紧的.作为应用,我们证明一个pinching结果,即在某些条件下,Ricci平坦的流形必定平坦.     

A Classification of Locally Homogeneous Affine Connections with Skew-Symmetric Ricci Tensor on 2-Dimensional Manifolds

 We classify, in an explicit form, the locally homogeneous torsionless affine connections as in the title. We also give some motivation for this research coming from the study of Osserman spaces. (Received 22 July 1999; in revised form 13 January 2000)     

李奇曲率平行的黎曼流形的曲率张量模长

李安民和赵国松［１］提出了下面的问题：找出李奇曲率平行的黎曼流形的曲率张量模长的最佳拼挤常数并确定达到该值的流形．本文确定了非爱因斯坦流形的最佳拼挤常数和达到该值的黎曼流形．在ｎ１２时，回答了［１］中提出的问题．     

On Almost Hermitian 4-manifolds with J-invariant Ricci Tensor

We prove that every almost Hermitian 4-manifold with J-invariant Ricci tensor which is conformally flat or has harmonic curvature is either a space of constant curvature or a Kähler manifold. We also obtain analogous results on almost Kähler 4-manifolds.     

Pseudo-Riemannian 3-manifolds with prescribed distinct constant Ricci eigenvalues

We study three-dimensional pseudo-Riemannian manifolds having distinct constant principal Ricci curvatures. These spaces are described via a system of differential equations, and a simple characterization is proved to hold for the locally homogeneous ones. We then generalize the technique used in [O. Kowalski, F. Prüfer, On Riemannian 3-manifolds with distinct constant Ricci eigenvalues, Math. Ann. 300 (1994) 17-28] for Riemannian manifolds and construct explicitly homogeneous and non-homogeneous pseudo-Riemannian metrics in R³, having the prescribed principal Ricci curvatures.     

Ricci Curvature Decay on Open Manifolds

The behaviour of the Ricci curvature along rays in a completeopen manifold is examined.     

Four-Dimensional Pseudo-Riemannian Generalized Symmetric Spaces Which are Algebraic Ricci Solitons

We classify, up to isometry, non-symmetric simply-connected four-dimensional pseudo-Riemannian generalized symmetric spaces which are algebraic Ricci solitons. It turns out that those of Cerny–Kowalski’s types A, C and D are algebraic Ricci solitons, whereas those of type B are not. Thus, we give new examples of algebraic Ricci solitons.     

Introduction to Actions of Discrete Groups on Pseudo-Riemannian Homogeneous Manifolds

Isometric actions of discrete groups are not always properly discontinuous for pseudo-Riemannian manifolds. This short exposition gives an up-to-date survey of some of the basic questions about discontinuous groups for pseudo-Riemannian homogeneous spaces, on which a rapid development has been made since late 1980s.The first half includes an elementary geometric motivation, the Calabi–Markus phenomenon, the discontinuous dual, and deformation. These topics are rebuilt on a criterion of properly discontinuous actions on homogeneous spaces of reductive groups, obtained by Kobayashi [Math. Ann. 1989] and generalized independently by Benoist [Ann. Math. 1996] and Kobayashi [J. Lie Theory 1996].The second half discusses the existence problem of compact Clifford–Klein forms of pseudo-Riemannian homogeneous spaces, for which many new methods from different areas have been recently employed. We examine these various approaches in some typical cases. We also point out that Zimmer's examples on SL(n)/SL(m) [J. Amer. Math. Soc. 1994] and Shalom's examples on SL(n)/SL(2) [Ann. Math. 2000] are readily obtained as special cases of Kobayashi's criterion [Duke Math. J. 1992], where the former uses ergodic theory and restrictions of unitary representations, respectively, while the latter uses cohomologies of discrete groups.The article also explains some open problems and conjectures.     

Bakry-■mery里奇曲率下的Calabi-型定理

得到了两个关于黎曼流形上Bakry-■mery里奇曲率沿着测地线的积分估计.作为应用,得到了两个Calabi定理的推广结果,即得到了流形是紧致的充分条件.     

Complete Open Manifolds with Nonnegative Ricci Curvature

In this paper, we study complete open manifolds with nonnegative Ricci curvature and injectivity radius bounded from below. We find that this kind of manifolds are diffeomorphic to a Euclidean space when certain distance functions satisfy a reasonable condition.     

Weighted Sobolev Inequalities and Ricci Flat Manifolds

In this paper, we prove a weighted Sobolev inequality and a Hardy inequality on manifolds with nonnegative Ricci curvature satisfying a reverse volume doubling condition. It enables us to obtain rigidity results for Ricci flat manifolds. Received: November 2006, Revision: April 2007, Accepted: April 2007