Showing papers on "Ricci decomposition published in 2000"

On the structure of spaces with Ricci curvature bounded below. I

Abstract: In this paper and in we study the structure of spaces Y which are pointed Gromov Hausdor limits of sequences f M i pi g of complete connected Riemannian manifolds whose Ricci curvatures have a de nite lower bound say RicMn i n In Sections and sometimes in we also assume a lower volume bound Vol B pi v In this case the sequence is said to be non collapsing If limi Vol B pi then the sequence is said to collapse It turns out that a convergent sequence is noncollapsing if and only if the limit has positive n dimensional Hausdor measure In par ticular any convergent sequence is either collapsing or noncollapsing Moreover if the sequence is collapsing it turns out that the Hausdor dimension of the limit is actually n see Sections and Our theorems on the in nitesimal structure of limit spaces have equivalent statements in terms of or implications for the structure on a small but de nite scale of manifolds with RicMn n Al though both contexts are signi cant for the most part it is the limit spaces which are emphasized here Typically the relation between corre sponding statements for manifolds and limit spaces follows directly from the continuity of the geometric quantities in question under Gromov Hausdor limits together with Gromov s compactness theorem Theorems see also Remark are examples of

The Classification of Spaces Defining Gravitational Fields

Abstract: In this paper written in 1954 Alexei Petrov describes his famous classification of spaces according to the algebraical structure of the curvature tensor, that determines the classes of the gravitational fields permitted therein. Now this classification of spaces (and, respectively, of the gravitational fields) is known as Petrov’s classification. This paper was originally published, in Russian, in Scientific Transactions of Kazan State University: Petrov A. Z. Klassifikazija prostranstv, opredelajuschikh polja tjagotenia. Uchenye Zapiski Kazanskogo Gosudarstvennogo Universiteta, 1954, vol. 114, book 8, pages 55–69. Translated from Russian in 2008 by Vladimir Yershov, England–Pulkovo. In this paper, the detailed proof of results obtained and published by the author earlier in 1951 [1]. Namely, it is shown that by examining the algebraic structure of the curvature tensor V4 one can establish a classification of the gravitational fields defined by this tensor and given in the form ds = gij dx dx , (1) with the fundamental tensor satisfying the field equations

Noncollapsing examples with positive Ricci curvature and infinite topological type

Abstract: Abstract. ((Without Abstract)).

Three-Dimensional Lorentz Metrics and Curvature Homogeneity of Order One

Abstract: In this paper we investigate the existence of three-dimensional Lorentzian manifolds which are curvature homogeneous up to order one but which are not locally homogeneous, and we obtain a complete local classification of these spaces. As a corollary we determine, for each Segre type of the Ricci curvature tensor, the smallest k ∈ N for which curvature homogeneity up to order k guarantees local homogeneity of the three-dimensional manifold under consideration.

Nonnegative Ricci curvature, small linear diameter growth and finite generation of fundamental groups

Abstract: In 1968, Milnor conjectured that a complete noncompact manifold with nonnegative Ricci curvature has a finitely generated fundamental group. The author applies the Excess Theorem of Abresch and Gromoll (1990), to prove two theorems. The first states that if such a manifold has small linear diameter growth then its fundamental group is finitely generated. The second states that if such a manifold has an infinitely generated fundamental group then it has a tangent cone at infinity which is not polar. A corollary of either theorem is the fact that if such a manifold has linear volume growth, then its fundamental group is finitely generated.

Generalized Taub-NUT metrics and Killing-Yano tensors

Abstract: The relation between “hidden” symmetries encapsulated in the Stackel-Killing tensors and the Killing-Yano tensors is investigated. A necessary condition that a Stackel-Killing tensor of valence 2 be the contracted product of a Killing-Yano tensor of valence 2 with itself is re-derived for a Riemannian manifold. This condition is applied to the generalized Euclidean Taub-NUT metrics which admit a Kepler type symmetry. It is shown that in general the Stackel-Killing tensors involved in the Runge-Lenz vector cannot be expressed as a product of Killing-Yano tensors. The only exception is the original Taub-NUT metric.

Almost Kähler 4-manifolds with J-invariant ricci tensor and special weyl tensor

Abstract: We study the question of integrability of a compatible almost complex structure on a compact symplectic 4-manifold, under various natural assumptions on the curvature of the associated almost Kahler metric.

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

Abstract: 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

Irreducible decompositions of physical tensors of high orders

Abstract: In this paper we use the term 'physical tensor' to stand for a tensor that belongs to a tensor subspace. Based on the relationship among the characters of rotation representation, some techniques are developed in order to give the numbers of independent deviatoric tensors contained in the irreducible decompositions of physical tensors, even prior to the constructions of the irreducible decompositions. A number of examples are shown and many of them are new results.

Symplectic 4-manifolds with Hermitian Weyl tensor

Abstract: It is proved that any compact almost Kihler, Einstein 4-manifold whose fundamental form is a root of the Weyl tensor is necessarily Kihler.

Integral geometry problems for symmetric tensor fields with incomplete data

Abstract: Abstract - The paper is devoted to the reconstruction problems of a compactly supported symmetric tensor field in ℝn , if its ray transform is known along the lines, intersecting a given curve. Two versions of the reconstruction are considered.

Symmetric symplectic spaces with Ricci-type curvature

Abstract: We determine the isomorphism classes of symmetric symplectic manifolds of dimension at least 4 which are connected, simply-connected and have a curvature tensor which has only one non-vanishing irreducible component — the Ricci tensor.

Harmonic functions on manifolds with nonnegative Ricci curvature and linear volume growth

Abstract: Lower bounds on Ricci curvature limit the volumes of sets and the existence of harmonic functions on Riemannian manifolds. In 1975, Shing Tung Yau proved that a complete noncompact manifold with nonnegative Ricci curvature has no nonconstant harmonic functions of sublinear growth. In the same paper, Yau used this result to prove that a complete noncompact manifold with nonnegative Ricci curvature has at least linear volume growth. In this paper, we prove the following theorem concerning harmonic functions on these manifolds. Theorem: Let M be a complete noncompact manifold with nonnegative Ricci curvature and at most linear volume growth. If there exists a nonconstant harmonic function, f, of polynomial growth of any given degree q, then the manifold splits isometrically, M= N x R.

Comparison of the Bondi-Sachs and Penrose approaches to asymptotic flatness

Abstract: Relations between the Bondi-Sachs approach and the Penrose conformal technique for asymptotically flat metrics are reviewed. Conditions on the conformal factor and the Ricci tensor are examined in order to compare the two approaches. The Bondi-Sachs coordinates are constructed (up to O( 3 )) for a class of Robinson-Trautman metrics. Some solutions within this class (with pure radiation fields) are given.

Integral pinching theorems

Abstract: Using Hamilton's Ricci flow we shall prove several pinching results for integral curvature. In particular, we show that if p>n/2$ and the Lp norm of the curvature tensor is small and the diameter is bounded, then the manifold is an infra-nilmanifold. We also obtain a result on deforming metrics to positive sectional curvature.

Ricci collineations for non-degenerate, diagonal and spherically symmetric ricci tensors.

Abstract: The expression of the vector field generator of a Ricci collineation for diagonal, spherically symmetric and non-degenerate Ricci tensors is obtained. The resulting expressions show that the time and radial first derivatives of the components of the Ricci tensor can be used to classify the collineation, leading to 64 families. Some examples illustrate how to obtain the collineation vector.

Conformal metrics and Ricci tensors in the pseudo-Euclidean space

Abstract: We consider constant symmetric tensors T on R', n > 3, and we study the problem of finding metrics j conformal to the pseudo-Euclidean metric g such that Ric g = T. We show that such tensors are determined by the diagonal elements and we obtain explicitly the metrics 9. As a consequence of these results we get solutions globally defined on Rn for the equation -So\g~So + njIVgsoI2/2 + AWo2 = 0. Moreover, we show that for certain unbounded functions K defined on Rn, there are metrics conformal to the pseudo-Euclidean metric with scalar curvature K.

Differential geometry of Grassmannian embeddings of based loop groups

Abstract: In this article we study differential geometric properties of the most basic infinite-dimensional manifolds arising from fermionic (1+1) -dimensional quantum field theory: the restricted Grassmannian and the group of based loops in a compact simple Lie group. We determine the Riemann curvature tensor and the (linearly) divergent expression corresponding to the Ricci curvature of the restricted Grassmannian after proving that the latter manifold is an isotropy irreducible Hermitian symmetric space. Using the Gauss equation of the embedding of a based loop group into the restricted Grassmannian we show that the (conditional) Ricci curvature of a based loop group is proportional to its metric. Furthermore we explicitly derive the logarithmically divergent behaviour of several differential geometric quantities arising from this embedding.

Classification of Ric-semiparallel hypersurfaces in Euclidean spaces

Abstract: A complete local classification and a geometric description are given for hypersurfaces with semiparallel Ricci tensor in Euclidean spaces.

On the projective geometry of sprays

Abstract: After a careful study of the mixed curvatures of the Berwald-type (in particular, Berwald) connections, we present an axiomatic description of the so-called Yano-type Finsler connections. Using the Yano connection, we derive an intrinsic expression of Douglas' famous projective curvature tensor and we also represent it in terms of the Berwald connection. Utilizing a clever observation of Z. Shen, we show in a coordinate-free manner that a “spray manifold” is projectively equivalent to an affinely connected manifold iff its Douglas tensor vanishes. From this result we infer immediately that the vanishing of the Douglas tensor implies that the projective Weyl tensor of the Berwald connection “depends only on the position”

Tensor products of irreducible representations of the groups gl(2,k) and sl(2,k) , k a finite field

Abstract: An important problem in Representation Theory consists in describing the irreducible representations of a group. In particular, since many times a group is presented as a direct product of some of its subgroups, the description of the tensor product of irreducible representations is a central issue. OIL the other hand, since induction from some subgroups has proved to be a very efficient tool for the theory, it is ow purpose to describe the tensor product of any two irreducible complex representations of the groups GL(2, k) and SL(2, k), k a finite field, essentially as induced representations of some of the tori of the above mentioned groups. Also, when one studies the problem of constructing models of a given group (Gelfand Models,Whittaker Models, ...) that i s to say, representations that contain with multiplicity free (sometimes all) the irreducible representations of the group, one sees the convenience of having the tensor product of irreducible (complex) representations as an induced representation. In fact, the study ol models for some classical groups will appear elsewhere. Let k be a finite field of q = pn elements, p prime and let K be the unique quadratic extension of k, generated by u E K. Let r be a non trivial element of the Galois group of K/k. We identify z E K with the k-automorphisni of K given by x +-+ zx. Furthermore, we denote by z the matrix of the above automorphism respect to the base (1, a) . In this way, the function z +-+ z E G, G = GL(2, k), defines a monomorphism of K in G whose image is the anisotropic torus T-l.

Estimation of Ricci tensor on certain Fano manifolds

Abstract: We give time existence to some Monge-Ampere equations on certain Fano manifolds which cannot carry Einstein-Kahler metrics. These solutions allow us to obtain an estimation of Ricci tensors on our manifolds.

A new singularity theorem in relativistic cosmology

Abstract: It is shown that if the timelike eigenvector of the Ricci tensor be hypersurface orthogonal so that the space time allows a foliation into space sections then the space average of each of the scalar that appear in the Raychaudhuri equation vanishes provided the strong energy condition holds good. This result is presented in the form of a singularity theorem.

Erratum to: “On the projective geometry of sprays” [Differential Geom. Appl. 12 (2000) 185–206]

Abstract: After a careful study of the mixed curvatures of the Berwald-type (in particular, Berwald) connections, we present an axiomatic description of the so-called Yano-type Finsler connections. Using the Yano connection, we derive an intrinsic expression of Douglas' famous projective curvature tensor and we also represent it in terms of the Berwald connection. Utilizing a clever observation of Z. Shen, we show in a coordinate-free manner that a “spray manifold” is projectively equivalent to an affinely connected manifold iff its Douglas tensor vanishes. From this result we infer immediately that the vanishing of the Douglas tensor implies that the projective Weyl tensor of the Berwald connection “depends only on the position”

On the Geometric Structure of Spatio-temporal Patterns

Abstract: The structure of hypersurfaces corresponding to different spatio-temporal patterns is considered, and in particular representations based on geometrical invariants, such as the Riemann and Einstein tensors and the scalar curvature are analyzed. The spatio-temporal patterns result from translations, Lie-group transformations, accelerated and discontinuous motions and modulations. Novel methods are obtained for the computation of motion parameters and the optical flow. Moreover, results obtained for accelerated and discontinuous motions are useful for the detection of motion boundaries.