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     

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.     

Pseudo-Riemannian weakly symmetric manifolds

There is a well-developed theory of weakly symmetric Riemannian manifolds. Here it is shown that several results in the Riemannian case are also valid for weakly symmetric pseudo-Riemannian manifolds, but some require additional hypotheses. The topics discussed are homogeneity, geodesic completeness, the geodesic orbit property, weak symmetries, and the structure of the nilradical of the isometry group. Also, we give a number of examples of weakly symmetric pseudo-Riemannian manifolds, some mirroring the Riemannian case and some indicating the problems in extending Riemannian results to weakly symmetric pseudo-Riemannian spaces.     

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.     

Pseudo-Riemannian manifolds with recurrent spinor fields

The existence of a recurrent spinor field on a pseudo-Riemannian spin manifold (M,g) is closely related to the existence of a parallel 1-dimensional complex subbundle of the spinor bundle of (M,g). We characterize the following simply connected pseudo-Riemannian manifolds that admit these subbundles in terms of their holonomy algebras: Riemannian manifolds, Lorentzian manifolds, pseudo-Riemannian manifolds with irreducible holonomy algebras, and pseudo-Riemannian manifolds of neutral signature admitting two complementary parallel isotropic distributions.     

Rough Isometry and <Emphasis Type="Italic">p</Emphasis>-Harmonic Boundaries of Complete Riemannian Manifolds

In this paper, we describe the behavior of bounded energy finite solutions for certain nonlinear elliptic operators on a complete Riemannian manifold in terms of its p-harmonic boundary. We also prove that if two complete Riemannian manifolds are roughly isometric to each other, then their p-harmonic boundaries are homeomorphic to each other. In the case, there is a one to one correspondence between the sets of bounded energy finite solutions on such manifolds. In particular, in the case of the Laplacian, it becomes a linear isomorphism between the spaces of bounded harmonic functions with finite Dirichlet integral on the manifolds. This work was supported by grant No. R06-2002-012-01001-0(2002) from the Basic Research Program of the Korea Science & Engineering Foundation.     

Compact pseudo-Riemannian manifolds with parallel Weyl tensor

It is shown that in every dimension n = 3j + 2, j = 1, 2, 3, . . ., there exist compact pseudo-Riemannian manifolds with parallel Weyl tensor, which are Ricci-recurrent, but neither conformally flat nor locally symmetric, and represent all indefinite metric signatures. The manifolds in question are diffeomorphic to nontrivial torus bundles over the circle. They all arise from a construction that a priori yields bundles over the circle, having as the fibre either a torus, or a 2-step nilmanifold with a complete flat torsionfree connection; our argument only realizes the torus case.     

Symplectic, complex and Kähler structures on four-dimensional generalized symmetric spaces

We obtain the full classification of invariant symplectic, (almost) complex and Kähler structures, together with their paracomplex analogues, on four-dimensional pseudo-Riemannian generalized symmetric spaces. We also apply these results to build some new examples of five-dimensional homogeneous K-contact, Sasakian, K-paracontact and para-Sasakian 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.     

Non-linear partial differential equations via vector fields on homogeneous Banach manifolds

The vector field formulation of and the Sato-Segal-Wilson approach to soliton equations are related to each other in this paper. From Banach Lie groups associated with the MKdV hierarchy of differential equations, we derive homogeneous Banach manifolds of solutions on which these equations are realized by vector fields. In the same way, we obtain homogeneous Banach manifolds of solutions to the sine-Gordon equation. The scattering and inverse scattering maps in this set-up are also discussed.     

Riemann extensions and affine differential geometry

In order to define an affine immersion of manifolds in affine differential geometry, it is necessary to choose a set of normal planes to the immersed manifold. The theory is then developed after this choice has been made. However, it was shown by A.G.Walker [WA1] that a torsion-free affine connexion on a manifold determines canonically a pseudo-Riemannian metric on the cotangent bundle, called the Riemann-extension of the affine connexion. By making use of this pseudo-Riemannian metric it is possible to define an affine immersion without making a suitable choice of normal planes.     

Poisson manifolds with compatible pseudo-metric and pseudo-Riemannian Lie algebras

In a previous paper (C. R. Acad. Sci. Paris Sér. I 333 (2001) 763–768), the author introduced a notion of compatibility between a Poisson structure and a pseudo-Riemannian metric. In this paper, we introduce a new class of Lie algebras called pseudo-Riemannian Lie algebras. The two notions are closely related: we prove that the dual of a Lie algebra endowed with its canonical linear Poisson structure carries a compatible pseudo-Riemannian metric if and only if the Lie algebra is a pseudo-Riemannian Lie algebra. Moreover, the Lie algebra obtained by linearizing at a point a Poisson manifold with a compatible pseudo-Riemannian metric is a pseudo-Riemannian Lie algebra. We also give some properties of the symplectic leaves of such manifolds, and we prove that every Poisson manifold with a compatible Riemannian metric is unimodular. Finally, we study Poisson Lie groups endowed with a compatible pseudo-Riemannian metric, and we give the classification of all pseudo-Riemannian Lie algebras of dimension 2 and 3.     

Sum of squares manifolds: The expressibility of the Laplace-Beltrami operator on pseudo-Riemannian manifolds as a sum of squares of vector fields

In this paper, we investigate under what circumstances the Laplace-Beltrami operator on a pseudo-Riemannian manifold can be written as a sum of squares of vector fields, as is naturally the case in Euclidean space.

We show that such an expression exists globally on one-dimensional manifolds and can be found at least locally on any analytic pseudo-Riemannian manifold of dimension greater than two. For two-dimensional manifolds this is possible if and only if the manifold is flat.

These results are achieved by formulating the problem as an exterior differential system and applying the Cartan-Kähler theorem to it.

     

Elliptic Operators on Manifolds with Singularities and K-Homology

Elliptic operators on smooth compact manifolds are classified by K-homology. We prove that a similar classification is valid also for manifolds with simplest singularities: isolated conical points and edges. The main ingredients of the proof of these results are: Atiyah–Singer difference construction in the noncommutative case and Poincaré isomorphism in K-theory for (our) singular manifolds. As an application we give a formula in topological terms for the obstruction to Fredholm problems on manifolds with edges.Mathematics Subject Classification (2000): 58J05(Primary), 19K33 35S35 47L15(Secondary)(Received: June 2004)     

K-Theoretic Indices of Dirac Type Operators on Complete Manifolds and the Roe Algebra

In this paper we study the K-theoretic indices of Dirac Type operators on complete manifolds and their geometric applications.     

Concircular vector fields on semi-Riemannian spaces

In this paper, we construct an analogue of concircular fields for semi-Riemannian spaces (i.e., for manifolds with degenerate metrics). We find a tensor criterion of spaces admitting the maximal number of concircular fields or having no such fields. We detect a gap in the distribution of dimensions of the space of concircular fields, which, in contrast to the corresponding gap in the case of pseudo-Riemannian manifolds, is lesser by 1. We also study some special types of concircular fields having no analogues for pseudo-Riemannian manifolds. The canonical form of the metric for some classes of semi-Riemannian spaces admitting concircular fields is obtained. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 31, Geometry, 2005.     

An equivariant Novikov conjecture

We discuss and formulate the correct equivariant generalization of the strong Novikov conjecture. This will be the statement that certain G-equivariant higher signatures (living in suitable equivariant K-groups) are invariant under G-maps of manifolds which, nonequivariantly, are homotopy equivalences preserving orientation. We prove this conjecture for manifolds modeled on a complete Riemannian manifold of nonpositive curvature on which G (a compact Lie group) acts by isometries. We also use the theory of harmonic maps to construct (in some cases) G-maps into such model spaces.Dedicated to Alexander GrothendieckPartially supported by NSF Grants DMS 84-00900 and 87-00551.Partially supported by NSF Grant DMS 86-02980, a Presidential Young Investigator Award, and a Sloan Foundation Fellowship.     

Manifolds which are Ivanov-Petrova or k -Stanilov

We present some examples of curvature homogeneous pseudo-Riemannian manifolds which are k-spacelike Jordan Stanilov.     

Covariant derivative of the curvature tensor of pseudo-Kählerian manifolds

It is well known that the curvature tensor of a pseudo-Riemannian manifold can be decomposed with respect to the pseudo-orthogonal group into the sum of the Weyl conformal curvature tensor, the traceless part of the Ricci tensor and of the scalar curvature. A similar decomposition with respect to the pseudo-unitary group exists on a pseudo-Kählerian manifold; instead of the Weyl tensor one obtains the Bochner tensor. In the present paper, the known decomposition with respect to the pseudo-orthogonal group of the covariant derivative of the curvature tensor of a pseudo-Riemannian manifold is refined. A decomposition with respect to the pseudo-unitary group of the covariant derivative of the curvature tensor for pseudo-Kählerian manifolds is obtained. This defines natural classes of spaces generalizing locally symmetric spaces and Einstein spaces. It is shown that the values of the covariant derivative of the curvature tensor for a non-locally symmetric pseudo-Riemannian manifold with an irreducible connected holonomy group different from the pseudo-orthogonal and pseudo-unitary groups belong to an irreducible module of the holonomy group.     

伪黎曼空间的2-调和类空子流形

讨论伪黎曼空间的2－调和类空子流形的全测地性及其与极大类空子流形的关系。