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.     

On Pseudo-Riemannian Manifolds Whose Ricci Tensor is Parallel

Ricci-parallel Riemannian manifolds have a diagonal Ricci endomorphism Ric and are therefore, at least locally, a product of Einstein manifolds. This fails in the pseudo-Riemannian case. Using, on the one side, a general result in linear algebra due to Klingenberg and on the other side, a theorem on the holonomy of pseudo-Riemannian manifolds, this work classifies the different types of pseudo-Riemannian manifolds whose Ricci tensor is parallel.     

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.     

Lightlike Hypersurfaces in Indefinite Trans-Sasakian Manifolds

This paper deals with lightlike hypersurfaces of indefinite trans-Sasakian manifolds of type (α, β), tangent to the structure vector field. Characterization Theorems on parallel vector fields, integrable distributions, minimal distributions, Ricci-semi symmetric, geodesibility of lightlike hypersurfaces are obtained. The geometric configuration of lightlike hypersurfaces is established. We prove, under some conditions, that there are no parallel and totally contact umbilical lightlike hypersurfaces of trans-Sasakian space forms, tangent to the structure vector field. We show that there exists a totally umbilical distribution in an Einstein parallel lightlike hypersurface which does not contain the structure vector field. We characterize the normal bundle along any totally contact umbilical leaf of an integrable screen distribution. We finally prove that the geometry of any leaf of an integrable distribution is closely related to the geometry of a normal bundle and its image under ${\overline{\phi}}$ .     

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.     

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.     

Slant Lightlike Submanifolds of Indefinite Cosymplectic Manifolds

In this paper, we introduce the notion of a slant lightlike submanifold of an indefinite Cosymplectic manifold. We provide a nontrivial example and obtain necessary and sufficient conditions for the existence of a slant lightlike submanifold. Also, we give an example of a minimal slant lightlike submanifold of R⁹₂{R^{9}_{2}} and prove some characterization Theorems.     

Symmetries of Null Geometry in Indefinite Kenmotsu Manifolds

Null hypersurfaces have metrics with vanishing determinants and this degeneracy of these metrics leads to several difficulties. In this paper, null hypersurfaces of indefinite Kenmotsu space forms, tangent to the structure vector field, are studied with specific attention to locally symmetric, semi-symmetric and Ricci semi-symmetric null hypersurfaces. We show that locally symmetric and semi-symmetric null hypersurfaces are totally geodesic and parallel. These also hold for Ricci semi-symmetric null hypersurfaces, under a certain condition. We prove that, in null Einstein hypersurfaces of an indefinite Kenmotsu space form, tangent to the structure vector field, the local symmetry, semisymmetry and Ricci semi-symmetry notions are equivalent. For totally contact umbilical null hypersurfaces, we show that there are η-“Weyl” connections adapted to the induced structure on the null hypersurface.     

Screen Pseudo-Slant Lightlike Submanifolds of Indefinite Sasakian Manifolds

In this paper, we introduce the notion of screen pseudo-slant lightlike submanifolds of indefinite Sasakian manifolds giving characterization theorem with some non-trivial examples of such submanifolds. Integrability conditions of distributions D ₁, D ₂ and RadTM on screen pseudo-slant lightlike submanifolds of indefinite Sasakian manifolds have been obtained. Further, we obtain necessary and sufficient conditions for foliations determined by above distributions to be totally geodesic. We also study mixed geodesic screen pseudo-slant lightlike submanifolds of indefinite Sasakian manifolds.     

Screen Integrable Lightlike Hypersurfaces of Indefinite Sasakian Manifolds

We investigate lightlike hypersurfaces of indefinite Sasakian manifolds, tangent to the structure vector field ξ and whose screen distribution is integrable. We prove some results on parallel vector fields and on a leaf of the integrable distribution of this class. A theorem on a geometrical configuration of the screen distribution is obtained. We show that any totally contact umbilical leaf of a screen integrable distribution of a lightlike hypersurface is an extrinsic sphere. Received: February 22, 2008., Revised: June 18, 2008., Accepted: July 10, 2008.     

Isometric Splitting for Actions of Simple Lie Groups on Pseudo-Riemannian Manifolds

We study the action of a connected noncompact simple Lie group G on a connected finite volume pseudo-Riemannian manifold M. Our main result provides a geometric splitting of the metric on such M that involves natural metrics on G. For suitable G such splitting is a warped product. For any G the splitting turns out to be a direct product in the ergodic case.     

The General Solution of the Eisenhart Equation and Projective Motions of Pseudo-Riemannian Manifolds

The solution of the Eisenhart equation for pseudo-Riemannian manifolds (Mⁿ,g) of arbitrary signature and any dimension is obtained. Thereby, pseudo-Riemannian h-spaces (i.e., spaces admitting nontrivial solutions h ≠ cg of the Eisenhart equation) of all possible types determined by the Segrè characteristic χ of the bilinear form h are found. Necessary and sufficient conditions for the existence of an infinitesimal projective transformation in (Mⁿ,g) are given. The curvature 2-form of a (rigid) h-space of type χ = {r₁, …, r_k} is calculated and necessary and sufficient conditions for this space to have constant curvature are obtained.

     

Pseudo-Riemannian geodesics and billiards

In pseudo-Riemannian geometry the spaces of space-like and time-like geodesics on a pseudo-Riemannian manifold have natural symplectic structures (just like in the Riemannian case), while the space of light-like geodesics has a natural contact structure. Furthermore, the space of all geodesics has a structure of a Jacobi manifold. We describe the geometry of these structures and their generalizations. We also introduce and study pseudo-Euclidean billiards, emphasizing their distinction from Euclidean ones. We present a pseudo-Euclidean version of the Clairaut theorem on geodesics on surfaces of revolution. We prove pseudo-Euclidean analogs of the Jacobi–Chasles theorems and show the integrability of the billiard in the ellipsoid and the geodesic flow on the ellipsoid in a pseudo-Euclidean space.     

Complete <Emphasis Type="Italic">k</Emphasis>-Curvature Homogeneous Pseudo-Riemannian Manifolds

Abstract. For k ≥ 2, we exhibit complete k-curvature homogeneous neutral signature pseudo-Riemannian manifolds which are not locally affine homogeneous (and hence not locally homogeneous). All the local scalarWeyl invariants of these manifolds vanish. These manifolds are Ricci flat, Osserman, and Ivanov-Petrova. Mathematics Subject Classification (2000): 53B20     

Equivariant Kasparov Theory and Generalized Homomorphisms

Let G be a locally compact group. We describe elements of KK ^G (A, B) by equivariant homomorphisms, following Cuntz's treatment in the non-equivariant case. This yields another proof for the universal property of KK ^G : It is the universal split exact stable homotopy functor. To describe a Kasparov triple (, , F) for A, B by an equivariant homomorphism, we have to arrange for the Fredholm operator F to be equivariant. This can be done if A is of the form ; and more generally if the group action on A is proper in the sense of Exel and Rieffel.     

On equivariant Kasparov theory and Spanier-Whitehead duality

Suppose thatG is a second countable compact Lie group and thatA andB are commutativeG-C*-algebras. Then the Kasparov groupKK _* ^G (A, B) is a bifunctor onG-spaces. It is computed here in terms of equivariant stable homotopy theory. This result is a consequence of a more general study of equivariant Spanier-Whitehead duality and uses in an essential way the extension of the Kasparov machinery to the setting of -G-C*-algebras. As a consequence, we show that if (X, x ₀) is a based separable compact metricG-ENR (such as a smooth compactG-manifold) and (Y, y ₀) is a based countableG-CW-complex then there is a natural isomorphism