Showing papers on "Ricci decomposition published in 2005"

Alignment and algebraically special tensors in lorentzian geometry

Abstract: We develop a dimension-independent theory of alignment in Lorentzian geometry, and apply it to the tensor classification problem for the Weyl and Ricci tensors. First, we show that the alignment condition is equivalent to the principal null direction equation. In 4 dimensions this recovers the usual Petrov types are recovered. For higher dimensions we prove that, in general, a Weyl tensor does not possess any aligned directions. We then go on to describe a number of additional algebraic types for the various alignment configurations. For the case of second-order symmetric (Ricci) tensors, we perform the classification by considering the geometric properties of the corresponding alignment variety.

Generalized gravity and a ghost

Abstract: We show that generalized gravity theories involving the curvature invariants of the Ricci tensor and the Riemann tensor as well as the Ricci scalar are equivalent to multi-scalar–tensor gravities with four-derivative terms. By expanding the action around a vacuum spacetime, the action is reduced to that of the Einstein gravity with four-derivative terms, and consequently there appears a massive spin-2 ghost in such generalized gravity theories in addition to a massive spin-0 field.

On kenmotsu manifolds

Abstract: The purpose of this paper is to study a Kenmotsu manifold which is derived from the almost contact Riemannian manifold with some special conditions. In general, we have some relations about semi-symmetric, Ricci semi-symmetric or Weyl semisymmetric conditions in Riemannian manifolds. In this paper, we partially classify the Kenmotsu manifold and consider the manifold admitting a transformation which keeps Riemannian curvature tensor and Ricci tensor invariant.

On the concircular curvature tensor of a contact metric manifold

Abstract: We classify N(•)-contact metric manifolds which sat- isfy Z(»;X) ¢ Z = 0, Z(»;X) ¢ R = 0 or R(»;X) ¢ Z = 0.

Curvature invariants of static spherically symmetric geometries

Abstract: We construct all independent local scalar monomials in the Riemann tensor at an arbitrary dimension, for the special regime of static spherically symmetric geometries. Compared to general spaces, their number is significantly reduced: the extreme example is the collapse of all invariants ~Weyl^k, to a single term at each k. The latter is equivalent to the Lovelock invariant L_k . Depopulation is less extreme for invariants involving rising numbers of Ricci tensors, and also depends on the dimension. The corresponding local gravitational actions and their solution spaces are discussed.

Double forms, curvature structures and the $(p,q)$-curvatures

Abstract: We introduce a natural extension of the metric tensor and the Hodge star operator to the algebra of double forms to study some aspects of the structure of this algebra. These properties are then used to study new Riemannian curvature invariants, called the (p, q)-curvatures. They are a generalization of the p-curvature obtained by substituting the Gauss-Kronecker tensor to the Riemann curvature tensor. In particular, for p = 0, the (0, q)-curvatures coincide with the H. Weyl curvature invariants, for p = 1 the (1, q)-curvatures are the curvatures of generalized Einstein tensors, and for q = 1 the (p, 1)-curvatures coincide with the p-curvatures. Also, we prove that the second H. Weyl curvature invariant is nonnegative for an Einstein manifold of dimension n > 4, and it is nonpositive for a conformally flat manifold with zero scalar curvature. A similar result is proved for the higher H. Weyl curvature invariants.

Decomposition of tensor products of modular irreducibles for SL2

Abstract: We use tilting modules to study the structure of the tensor product of two simple modules for the algebraic group $\\SL_2$, in positive characteristic, obtaining a twisted tensor product theorem for its indecomposable direct summands. Various other related results are obtained, and numerous examples are computed.

Tensor products of psl(2|2) representations

Abstract: The aim of this work is to study finite dimensional representations of the Lie superalgebra psl(2|2) and their tensor products In particular, we shall decompose all tensor products involving typical (long) and atypical (short) representations as well as their so-called projective covers While tensor products of long multiplets and projective covers close among themselves, we shall find an infinite family of new indecomposables in the tensor products of two short multiplets Our note concludes with a few remarks on possible applications to the construction of AdS_3 backgrounds in string theory

Regularity of ghosts in tensor tomography

Abstract: We study on a compact Riemannian manifold with boundary the ray transform I which integrates symmetric tensor fields over geodesics. A tensor field is said to be a nontrivial ghost if it is in the kernel of I and is L2-orthogonal to all potential fields. We prove that a nontrivial ghost is smooth in the case of a simple metric. This implies that the wave front set of the solenoidal part of a field f can be recovered from the ray transform If. We give an explicit procedure for recovering the wave front set.

A Weyl-covariant tensor calculus

Abstract: On a (pseudo-) Riemannian manifold of dimension n⩾3, the space of tensors which transform covariantly under Weyl rescalings of the metric is built. This construction is related to a Weyl-covariant operator D whose commutator [D,D] gives the conformally invariant Weyl tensor plus the Cotton tensor. So-called generalized connections and their transformation laws under diffeomorphisms and Weyl rescalings are also derived. These results are obtained by application of Becchi Rouet Stora Tyutin techniques.

Congruences of Fluids in a Finslerian Anisotropic Space-Time

Abstract: We derive the generalized Raychauduri equation concepts of expansion, shear and vorticity. We give the Ricci tensor of a constant-curvature Randers–Finsler space metric whose first term is the Robertson–Walker metric.

A pinching estimate for solutions of the linearized Ricci flow system on 3-manifolds

Abstract: An important component of Hamilton’s program for the Ricci flow on compact 3-manifolds is the classification of singularities which form under the flow for certain initial metrics. In particular, Type I singularities, where the evolving metrics have curvatures whose maximums are inversely proportional to the time to blow-up, are modelled on the 3-sphere and the cylinder S × R and their quotients. On the other hand, Type II singularities (the complementary case) are much more difficult to understand. Despite this, it is known from the work of Hamilton that their singularity models are stationary solutions to the Ricci flow. This uses several techniques, including Harnack inequalities of Li-Yau-Hamilton type, the strong maximum principle for systems, dimension reduction, and the study of the geometry at infinity of noncompact stationary solutions (see§§1426 of [ H2].) In terms of Hamilton’s program, at least two obstacles remain: obtaining an injectivity radius estimate for Type II solutions and ruling out the so-called cigar soliton (the unique complete stationary solution on a surface with positive curvature) as the dimension reduction of a Type II singularity model. ∗Research partially supported by NSF grant DMS-9971891. 1In fact, Hamilton and Yau have announced informally that these are the only two obstacles and that they both would follow from obtaining a suitable differential matrix Harnack inequality of Li-Yau-Hamilton type for arbitrary solutions of the Ricci flow on compact 3manifolds. 2Added in proof: Very recently, Perelman [P1] has given a proof of the Little Loop Conjecture in all dimensions without curvature restriction. See also [P2] for some further developments.

Almost Kähler 4-dimensional Lie groups with J-invariant Ricci tensor☆

Abstract: The J -invariance of the Ricci tensor is a natural weakening of the Einstein condition in almost Hermitian geometry. The aim of this paper is to determine left-invariant strictly almost Kahler structures ( g , J , Ω ) on real 4-dimensional Lie groups such that the Ricci tensor is J -invariant. We prove that all these Lie groups are isometric (up to homothety) to the (unique) 4-dimensional proper 3-symmetric space.

Porous media equation and Sobolev inequalities under negative curvature

Abstract: We study the link between some modified porous media equation and Sobolev inequalities on a Riemannian manifold M whose Ricci curvature tensor is bounded below by a negative constant −ρ. The method used deals with entropy–energy differentiation and follows the way the author got inequalities under nonnegative Ricci curvature assumptions. The key of the proof is the curvature-dimension criterion.

Conformal entropy for generalized gravity theories as a consequence of horizon properties

Abstract: We show that a microscopic entropy formula based on Virasoro algebra follows from properties of stationary Killing horizons for Lagrangians with arbitrary dependence on Riemann tensor. The properties used are a consequence of regularity of invariants of Riemann tensor on the horizon. Eventual generalization of these results to Lagrangians with derivatives of Riemann tensor, as suggested by an example treated in the paper, relies on assuming regularity of invariants involving derivatives of Riemann tensor. This assumption however leads also to new interesting restrictions on metric functions near the horizon.

A note on the peeling theorem in higher dimensions

Abstract: We demonstrate the ``peeling property'' of the Weyl tensor in higher dimensions in the case of even dimensions (and with some additional assumptions), thereby providing a first step towards understanding of the general peeling behaviour of the Weyl tensor, and the asymptotic structure at null infinity, in higher dimensions.

On some class of hypersurfaces with three distinct principal curvatures

Abstract: ON SOME CLASS OF HYPERSURFACES WITH THREE DISTINCT PRINCIPAL CURVATURES KATARZYNA SAWICZ Institute of E onometri s and Computer S ien e Te hni al University of Cz sto howa Armii Krajowej 19B, 42-200 Cz sto howa, Poland E-mail: ksawi z zim.p z. zest.pl Abstra t. We investigate hypersurfa es M in spa es of onstant urvature with some spe ial minimal polynomial of the se ond fundamental tensor H of third degree. We present a urvature hara terization of pseudosymmetry type for su h hypersurfa es. We also prove that if su h a hypersurfa e is a manifold with pseudosymmetri Weyl tensor then it must be pseudosymmetri . 1. Introdu tion. Let M , n = dimM ≥ 3, be a onne ted hypersurfa e in a semiRiemannian manifold (N, g ). We denote by g the metri tensor indu ed on M from g . Further, let H, resp. A, be the se ond fundamental tensor, resp. the shape operator, of (M, g) in (N, g ). It is well known that H(X,Y ) = g(AX,Y ), for any ve tor elds X and Y tangent to M . We de ne the (0, 2)-tensor Hk, k ≥ 1, by H(X,Y ) = g(AX,Y ), where H = H and A = A. In Se tions 3 and 4 we present further basi fa ts relating to hypersurfa es. A hypersurfa e M , n ≥ 3, in (N, g ) is said to be quasi-umbili al at x ∈M if at this point we have H = αg + βu⊗ u, u ∈ T ∗ xM, α, β ∈ R. (1) If α = 0 (resp., β = 0 or α = β = 0) at x then M is alled ylindri al (resp., umbili al or geodesi ) at x. If (1) is ful lled at every point of M then it is alled a quasi-umbili al hypersurfa e. A hypersurfa e M , n ≥ 4, in (N, g ) is said to be 2-quasi-umbili al at x ∈M (see [16℄ and referen es therein) if at this point we have H = αg + βu⊗ u+ γv ⊗ v, u, v ∈ T ∗ xM, α, β, γ ∈ R, (2) 2000 Mathemati s Subje t Classi ation: Primary 53B20, 53B25; Se ondary 53C25.

On c-bochner curvature tensor of a contact metric manifold

Abstract: We prove that a (•;")-manifold with vanishing E- Bochner curvature tensor is a Sasakian manifold. Several inter- esting corollaries of this result are drawn. Non-Sasakian (•;")- manifolds with C-Bochner curvature tensor B satisfying B (»;X) ¢ S = 0, where S is the Ricci tensor, are classifled. N(•)-contact met- ric manifolds M 2n+1 , satisfying B (»;X)¢R = 0 or B (»;X)¢B = 0 are classifled and studied.

Exact and Approximate Quadratures for Curvature Tensor Estimation

Abstract: Accurate estimations of geometric properties of a surface from its discrete approximation are important for many computer graphics and geometric modeling applications. In this paper, we derive exact quadrature formulae for mean curvature, Gaussian curvature, and the Taubin integral representation of the curvature tensor. The exact quadratures are then used to obtain reliable estimates of the curvature tensor of a smooth surface approximated by a dense triangle mesh. The proposed method is fast and easy to implement. It is highly competitive with conventional curvature tensor estimation approaches. Additionally, we show that the curvature tensor approximated as proposed by us converges towards the true curvature tensor as the edge lengths tend to zero.

Geometric quantities of manifolds with Grassmann structure

Abstract: We study geometry of manifolds endowed with a Grassmann structure which depends on symmetries of their curvature. Due to this reason a complete decomposition of the space of curvature tensors over tensor product vector spaces into simple modules under the action of the group G = GL(p, ℝ) ⊗ GL(q, ℝ) is given. The dimensions of the simple submodules, the highest weights and some projections are determined. New torsion-free connections on Grassmann manifolds apart from previously known examples are given. We use algebraic results to reveal obstructions to the existence of corresponding connections compatible with some type of normalizations and to enlighten previously known results from another point of view.

Weyl manifolds with semi-symmetric connection

Abstract: We define a semi-symmetric connection on a Weyl manifold and study projective curvature tensor and conformal curvature tensor after giving some properties of the curvature tensor with respect to semi-symmetric connection.

The bach tensor and other divergence-free tensors

Abstract: In four dimensions, we prove that the Bach tensor is the only symmetric divergence-free 2-tensor which is also quadratic in Riemann and has good conformal behavior. In n > 4 dimensions, we prove that there are no symmetric divergence-free 2-tensors which are also quadratic in Riemann and have good conformal behavior, nor are there any symmetric divergence-free 2-tensors which are concomitants of the metric tensor gab together with its first two derivatives, and have good conformal behavior.

The First Eigenvalue of a Compact Manifold

Abstract: This paper discusses the first eigenvalue on a compact Riemann manifold with the negative lower bound Ricci curvature. Let M be a compact Riemann manifold with the Ricci curvature≥-R, R=const.≥0 and d is the diameter of M. Our main result is that the first eigenvalue λ1 of M satisfies λ1≥π2/d2-0.518.R.

Schouten tensor equations in conformal geometry with prescribed boundary metric.

Abstract: On a manifold with boundary, we deform the metric conformally. This induces a deformation of the Schouten tensor. We fix the metric at the boundary and realize a prescribed value for the product of the eigenvalues of the Schouten tensor in the interior, provided that there exists a subsolution.

Parallel transports in tensor spaces generated by derivations of tensor algebras

Abstract: The (parallel linear) transports in tensor spaces generated by derivations of the tensor algebra along paths are axiomatically described. Certain their properties are investigated. Transports along paths defined by derivations of the tensor algebra over a differentiable manifold are considered.

On a Normal Contact Metric Manifold

Abstract: We find the expression of the curvature tensor field for a manifold with is endowed with an almost contact structure satisfying the condition (1.7). By using this condition we obtain some properties of the Ricci tensor and scalar curvature (d. Theorem 3.2 and Proposition 3.2).

Tensor products of direct sums

Abstract: A similar formula to the one established by Ansemil and Floret for symmetric tensor products of direct sums is proved for alternating and Jacobian tensor products. It is then applied to stable spaces where a number of isomorphisms between spaces of tensors or multilinear forms are unveiled. A connection between these problems and irreducible group representations is made.