Locally conformally homogeneous pseudo-Riemannian spaces

Locally homogeneous Riemannian spaces were studied in [1–4]. Locally conformally homogeneous Riemannian spaces were considered in [10]. Moreover, the theorem claiming that every such space is either conformally flat or conformally equivalent to a locally homogeneous Riemannian space was proved.In this article, we study locally conformally homogeneous pseudo-Riemannian spaces and prove a theorem on their structure. Using three-dimensional Lie groups and the six-dimensional Heisenberg group [11], we construct some examples showing the difference between the Riemannian and pseudo-Riemannian cases for such spaces.     

On the structure of pseudo-Riemannian symmetric spaces

Following our approach to metric Lie algebras developed in a previous paper we propose a way of understanding pseudo-Riemannian symmetric spaces which are not semisimple. We introduce cohomology sets (called quadratic cohomology) associated with orthogonal modules of Lie algebras with involution. Then we construct a functorial assignment which sends a pseudo-Riemannian symmetric space M to a triple consisting of:

Generalized pseudo-Riemannian geometry

Generalized tensor analysis in the sense of Colombeau's construction is employed to introduce a nonlinear distributional pseudo-Riemannian geometry. In particular, after deriving several characterizations of invertibility in the algebra of generalized functions, we define the notions of generalized pseudo-Riemannian metric, generalized connection and generalized curvature tensor. We prove a ``Fundamental Lemma of (pseudo-) Riemannian geometry' in this setting and define the notion of geodesics of a generalized metric. Finally, we present applications of the resulting theory to general relativity.

     

On the geometry of stable discontinuous subgroups acting on threadlike homogeneous spaces

Following the notion of stability introduced by T. Kobayashi and S. Nasrin in [14], we show in the context of a threadlike Lie group G that any non-Abelian discrete subgroup is stable. One consequence is that any resulting deformation space ?(Γ,G,H) is a Hausdorff space, where Γ acts on the threadlike homogeneous space G/H as a discontinuous subgroup. Whenever k = rank(Γ) > 3, this space is also shown to be endowed with a smooth manifold structure. But if k = 3, then ?(Γ,G,H) admits a smooth manifold structure as its open dense subset. These phenomena are strongly linked to the features of adjoint orbits of the basis group G on the parameter space ?(Γ,G,H) (which is semi-algebraic in this case) and specifically to their dimensions, as it will be seen throughout the paper. This also allows to provide a proof of the Local Rigidity Conjecture in this setup.     

Flat homogeneous pseudo-Riemannian manifolds

The complete homogeneous pseudo-Riemannian manifolds of constant non-zero curvature were classified up to isometry in 1961 [1]. In the same year, a structure theory [2] was developed for complete flat homogeneous pseudo-Riemannian manifolds. Here that structure theory is sharpened to a classification. This completes the classification of complete homogeneous pseudo-Riemannian manifolds of arbitrary constant curvature.Research partially supported by N.S.F. Grant DMS 93 21285.     

Reductive homogeneous pseudo-Riemannian manifolds

A classification of homogeneous pseudo-Riemannian structures and a characterization of each primitive class are obtained. Several examples are also given.     

Quantum homogeneous spaces with faithfully flat module structures

LetA be a Hopf algebra with bijective antipode andB⊃A a right coideal subalgebra ofA. Formally, the inclusionB⊃A defines a quotient mapG→X whereG is a quantum group andX a right homogeneousG-space. From an algebraic point of view theG-spaceX only has good properties ifA is left (or right) faithfully flat as a module overB. In the last few years many interesting examples of quantumG-spaces for concrete quantum groupsG have been constructured by Podleś, Noumi, Dijkhuizen and others (as analogs of classical compact symmetric spaces). In these examplesB consists of infinitesimal invariants of the function algebraA of the quantum group. As a consequence of a general theorem we show that in all these casesA as a left or rightB-module is faithfully flat. Moreover, the coalgebraA/AB ⁺ is cosemisimple.     

Twisted products in pseudo-Riemannian geometry

Twisted products are generalizations of warped products, namely the warping function may depend on the points of both factors. The two canonical foliations of a twisted product are mutually perpendicular and their leaves are totally geodesic, resp. totally umbilic. The main result is a decomposition theorem of de Rham type: If on a simply connected, geodesically complete pseudo-Riemannian manifoldM two foliations with the above properties are given, thenM is a twisted product.     

On the product of homogeneous spaces

Within the class of Tychonoff spaces, and within the class of topological groups, most of the natural questions concerning ‘productive closure’ of the subclasses of countably compact and pseudocompact spaces are answered by the following three well-known results: (1) [ZFC] There is a countably compact Tychonoff space X such that X × X is not pseudocompact; (2) [ZFC] The product of any set of pseudocompact topological groups is pseudocompact; and (3) [ZFC+ MA] There are countably compact topological groups G₀, G₁ such that G₀ × G₁ is not countably compact.In this paper we consider the question of ‘productive closure” in the intermediate class of homogeneous spaces. Our principal result, whose proof leans heavily on a simple, elegant result of V.V. Uspenski?, is this: In ZFC there are pseudocompact, homogeneous spaces X₀, X₁ such that X₀ × X₁ is not pseudocompact; if in addition MA is assumed, the spaces X_i may be chosen countably compact.Our construction yields an unexpected corollary in a different direction: Every compact space embeds as a retract in a countably compact, homogeneous space. Thus for every cardinal number α there is a countably compact, homogeneous space whose Souslin number exceeds α.     

On the signature of homogeneous spaces

We compute the signature of real and quaternionic Grassmannians, thereby completing the table of signatures of symmetric spaces given in a previous paper [4]. In addition, all homogeneous spaces of exceptional Lie groups with non-zero signature are listed.     

On the geometry of moduli spaces

We construct a Kähler metric on the moduli spaces of compact complex manifolds with c₁,<0 and of polarized compact Kähler manifolds with c₁=0, which is a generalization of the Petersson-Well metric. It is induced by the variation of the Kähler-Einstein metrics on the fibers that exist according to the Calabi-Yau theorem. We compute the above metric on the moduli spaces of polarized tori and symplectic manifolds. It turns out to be the Maaß metric on the Siegel upper half space and the Bergmann metric on a symmetric space of type III resp. In particular it is Kähler-Einstein with negative curvature.Dedicated to Karl SteinHeisenberg-Stipendiat der Deutschen Forschungsgemeinschaft     

On the geometry of Kawaguchi spaces

In the paper, with the use of the E. Cartan exterior forms method, the theory of linear and affine connections of the generalized Kawaguchi space of order two is constructed. It is proved that the linear connection of this space incorporates intrinsic antiquaternional structures, the conditions of their complete integrability are found, and the affine connections associated with the above-mentioned structures are constructed. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 321–334, July–September, 2000. Translated by R. Lapinskas     

The geometry generated on homogeneous spaces by morphisms ofG-spaces

This survey is a continuation of the survey published in the Problems of Geometry Series, Vol. 15. It consists of an exposition of the general theory of homogeneous spaces defined by prescribing a set of endomorphisms of a Lie groupG. The morphisms of the homogeneous spaces so constructed are built up in the standard way and the geometry generated by these morphisms is studied.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 19, pp. 155–185, 1987.     

Circles and spheres in pseudo-Riemannian geometry

Summary A totally umbilical pseudo-Riemannian submanifold with the parallel mean curvature vector field is said to be an extrinsic sphere. A regular curve in a pseudo-Riemannian manifold is called a circle if it is an extrinsic sphere. LetM be ann-dimensional pseudo-Riemannian submanifold of index (0n) in a pseudo-Riemannian manifold with the metricg and the second fundamental formB. The following theorems are proved. For ₀ = +1 or –1, ₁ = +1, –1 or 0 (2–2 ₀+ ₁2n–2–2) and a positive constantk, every circlec inM withg(c, c) = ₀ andg( _c c, _c c) = ₁ k ² is a circle in iffM is an extrinsic sphere. For ₀ = +1 or –1 (–₀n–), every geodesicc inM withg(c, c) = ₀ is a circle in iffM is constant isotropic and B(x,x,x) = 0 for anyx T(M). In this theorem, assume, moreover, that 1n–1 and the first normal space is definite or zero at every point. Then we can prove thatM is an extrinsic sphere. When = 0 orn, this fact does not hold in general.