Showing papers on "Ricci decomposition published in 1997"

Ricci curvature and volume convergence

Abstract: The purpose of this paper is to give a new (integral) estimate of distances and angles on manifolds with a given lower Ricci curvature bound. This will provide us with an integral version of the Toponogov comparison triangle theorem for Ricci curvature and "almost extreme triangles" (see the earlier works [Cl] and [C2] for an analog of this when the manifold has positive Ricci curvature). Using this, we prove the Anderson-Cheeger conjecture saying that the volume is a continuous function on the space of all closed n-manifolds with Ricci curvature greater or equal to -(n - 1) equipped with the GromovHausdorff metric. We also prove Gromov's conjecture (for n 57 3) saying that an almost nonnegatively Ricci curved n-manifold with first Betti number equal to n is a torus. Further, we prove a conjecture of Anderson-Cheeger saying that an open n-manifold with nonnegative Ricci curvature whose tangent cone at infinity is in is itself in. Finally we prove a conjecture of Fukaya-Yamaguchi. We will now describe these results in more detail. Let dGH denote the Gromov-Hausdorff distance [GLP]. First we have the following result which was conjectured by Anderson-Cheeger.

Some Properties of the Bel and Bel± Robinson Tensors

Abstract: The properties of the Bel and Bel-Robinson tensors seem to indicate that they are closely related to the gravitational energy-momentum We present some new properties of these tensors which might throw some light onto this relationship First, for any spacetime we find a decomposition of the Bel tensor in terms of the Bel-Robinson tensor and two other tensors, which we call the “pure matter” super-energy tensor and the “matter-gravity coupling” super-energy tensor We show that the pure matter super-energy tensor of any Einstein-Maxwell field is simply the “square” of the usual energy-momentum tensor This, together with the fact that the Bel-Robinson tensor has dimensions of energy density square, leads us to the definition of square root for the Bel-Robinson tensor: a two-covariant symmetric traceless tensor with dimensions of energy density and such that its “square” gives the Bel-Robinson tensor We prove that this square root exists if and only if the spacetime is of Petrov type O, N or D, and its general expression is explicitly presented The properties of this new tensor are examined and some interesting explicit examples are analyzed Of particular interest are an invariant function that appears in the spherically symmetric metrics and an expression for the energy carried out by pure plane gravitational waves We also examine the decomposition of the whole Bel tensor for Vaidya's radiating metric and Kerr-Newman's solution Finally, we generalize the definition of square root to a factorization of the Bel-Robinson tensor and get the general solution for all Petrov types

Direct ζ-function approach and renormalization of one-loop stress tensors in curved spacetimes

Abstract: A method which uses a generalized tensorial $\zeta$-function to compute the renormalized stress tensor of a quantum field propagating in a (static) curved background is presented. The starting point of the method is the direct computation of the functional derivatives of the Euclidean one-loop effective action with respect to the background metric. This method, when available, gives rise to a conserved stress tensor and produces the conformal anomaly formula directly. It is proven that the obtained stress tensor agrees with statistical mechanics in the case of a finite temperature theory. The renormalization procedure is controlled by the structure of the poles of the stress-tensor $\zeta$ function. The infinite renormalization is automatic and is due to a ``magic'' cancellation of two poles. The remaining finite renormalization involves conserved geometrical terms arising by a certain residue. Such terms renormalize coupling constants of the geometric part of Einstein's equations (customary generalized through high-order curvature terms). The method is checked on particular cases (closed and open Einstein`s universe) finding agreement with other approaches. The method is also checked considering a massless scalar field in the presence of a conical singularity in the Euclidean manifold (i.e. Rindler spacetimes/large mass black hole manifold/cosmic string manifold). There, the method gives rise to the stress tensor already got by the point-splitting approach for every coupling with the curvature regardless of the presence of the singular curvature. Comments on the measure employed in the path integral, the use of the optical manifold and different approaches to renormalize the Hamiltonian are made.

Stability of the p-Curvature Positivity under Surgeries and Manifolds with Positive Einstein Tensor

Abstract: We establish the stability of the class of manifolds with positive p-curvature under surgeries in codimension ≥ p + 3. As a consequence of this result, we first obtain the classification of compact 2-connected manifolds of dimension ≥ 7 with positive Einstein tensor; and secondly the existence of metrics with positive Einstein tensor on any compact, simply connected, non-spin manifold of dimension ≥ 7 whose second homotopy group is isomorphic to Z2.

The spectrum of the Laplacian on a manifold of nonnegative Ricci curvature

Abstract: The study of the spectrum of the Laplacian on a complete noncompact Riemannian manifold has received much attention during the past decade or so. In particular, it has been conjectured and partially verified that the spectrum of the Laplacian acting on the space of L functions (the L spectrum) is the half line [0,∞) when the underlying manifold has nonnegative Ricci curvature. J. F. Escobar and A. Freire ([E], [E-F]) have dealt with the case that the manifold has nonnegative sectional curvature and proved among other things that this is true provided that the exponential map from the normal bundle of a soul of the manifold is a diffeomorphism onto the manifold. Later, J. Li in [Li] showed that this is also true for a Ricci nonnegative manifold possessing a pole, namely, the exponential map at some point is a diffeomorphism from the tangent space onto the manifold. Recently, the author was informed that H. Donnelly has considered the Ricci nonnegative manifolds with maximal volume growth. Among other things, he established also that the L spectrum of the Laplacian is [0,∞) in this case. This last result has also been obtained independently by N. Castañeda in [C]. In this short note, our purpose is to demonstrate the validity of the conjecture without imposing extra assumptions on the manifold. In fact, we shall prove more generally that the L spectrum of the Laplacian on a complete noncompact Riemannian manifold with asymptotically nonnegative Ricci curvature is given by the half line [0,∞). Recall that a complete manifold (M, g) has asymptotically nonnegative Ricci curvature if there exists a small constant δ(n) > 0 depending only on n such that for some point q ∈ M , the Ricci curvature satisfies RicM (x) ≥ −δ(n)r−2(x), where r(x), the distance from x to q, is sufficiently large. Note that here we do not address the problem whether there is any eigenvalue for the Laplacian.

Tensor products of analytic continuations of holomorphic discrete series

Abstract: We give the irreducible decomposition of the tensor product of an an- alytic continuation of the holomorphic discrete series of SU(2, 2) with its conjugate. 0. Introduction. The work of Segal (IES) and Mautner (M) established the abstract Plancherel theorem for type I groups. This meant that for an arbitrary unitary represen- tation, one could find its spectral decomposition into irreducibles and a corresponding spectral measure. To make this program explicit on L 2 -spaces on homogeneous spaces is one of the main subjects of harmonic analysis. Another interesting case is that of decom- posing a tensor product of irreducible representations; our aim in this paper is to consider this for certain holomorphic representations. The problem of finding the irreducible decomposition of tensor products of holomor- phic discrete series of the group SL(2, ) has been studied by Repka (Re1). The results there were used by Howe (How) to give the decomposition of the metaplectic represen- tation for certain dual pairs. See also (OZ). For a general semisimple Lie group G of Hermitian type a similar problem is studied in (Re2). It is shown that the tensor prod- uct of a scalar holomorphic discrete series with its conjugate is unitarily equivalent to the L 2 -space on the corresponding Hermitian symmetric space, L 2 (G K). Therefore we know its decomposition from the known theory of Harish-Chandra; namely L 2 (G K) W H ( ) C( ) 2 d

Harish-Chandra vertices and Steinberg's tensor product theorems for finite general linear groups

Abstract: Representations of Hecke and in non-describing characteristics. 1991 Mathematics Subject Classification: 20C20, 20C33, 20G05, 20G40.

Constraints on singularities under Ricci curvature bounds

Abstract: We announce results giving constraints on the singularities of spaces which are Gromov-Hausdorf'f limits of sequences of Riemannian manifolds whose Ricci curvature and volume are bounded from below and whose curvature tensor is bounded in an integral sense.

Tensor Distributions in the Presence of Degenerate Metrics

Abstract: Tensor distributions and their derivatives are described without assuming the presence of a metric This provides a natural framework for discussing tensor distributions on manifolds with degenerate metrics, including in particular metrics which change signature

Irreducible decomposition for tensor prodect representations of Jordanian quantum algebras

Abstract: Tensor products of irreducible representations of the Jordanian quantum algebras U_h(sl(2)) and U_h(su(1,1)) are considered. For both the highest weight finite dimensional representations of U_h(sl(2)) and lowest weight infinite dimensional ones of U_h(su(1,1)), it is shown that tensor product representations are reducible and that the decomposition rules to irreducible representations are exactly the same as those of corresponding Lie algebras.

Ricci curvature and betti numbers

Abstract: We derive a uniform bound for the total betti number of a closed manifold in terms of a Ricci curvature lower bound, a conjugate radius lower bound and a diameter upper bound. The result is based on an angle version of Toponogov comparison estimate for small triangles in a complete manifold with a Ricci curvature lower bound. We also give a uniform estimate on the generators of the fundamental group and prove a fibration theorem in this setting.

Some Results on Contact Metric Manifolds

Abstract: If the sectional curvatures of plane sections containing the characteristic vector field of a contact metric manifold M are non-vanishing, then we prove that a second order parallel tensor on M is a constant multiple of the associated metric tensor. Next, we prove for a contact metric manifold of dimension greater than 3 and whose Ricci operator commutes with the fundamental collineation that, if its Weyl conformal tensor is harmonic, then it is Einstein. We also prove that, if the Lie derivative of the fundamental collineation along the characteristic vector field on a contact metric 3-manifold M satisfies a cyclic condition, then M is either Sasakian or locally isometric to certain canonical Lie-groups with a left invariant metric. Next, we prove that if a three-dimensional Sasakian manifold admits a non-Killing projective vector field, it is of constant curvature 1. Finally, we prove that a conformally recurrent Sasakian manifold is locally isometric to a unit sphere.

A Use of Ideal Decomposition in the Computer Algebra of Tensor Expressions

Abstract: Let I be a left ideal of a group ring C[G] of a finite group C, for which a decomposition I = e;-1 Ik into minimal left ideals IA; is given. We present an algorithm, which determines a decomposition of the left ideal I . a, a E C[G], into minimal left ideals and a corresponding set of primitive orthogonal idempotents by means of a computer. The algorithm is motivated by the computer algebra of tensor expressions. Several aspects of the connection between left ideals of the group ring C[S,.] of a symmetric group 8,., their decomposition and the reduction of tensor expressions are discussed.

On a group approach to studying the Einstein and maxwell equations

Abstract: Based on the classical problem for decomposition of the tensor product of representations into irreducible components, which is considered in the elementary representation theory for orthogonal groups, a partial classification of the Einstein equations is carried out. A new class of Maxwell equations for relativistic electrodynamics is singled out and studied. Pointwise-irreducible decompositions for the energy-momentum and electromagnetic field tensors are obtained and a physical interpretation of the decomposition components is given.

On the ricci tensor of a real hypersurface of quaternionic hyperbolic space

Abstract: We prove the non-existence of Einstein real hypersurfaces of quaternionic hyperbolic space.

Graded metrics adapted to splittings

Abstract: Homogeneous graded metrics over split ℤ2-graded manifolds whose Levi-Civita connection is adapted to a given splitting, in the sense recently introduced by Koszul, are completely described. A subclass of such is singled out by the vanishing of certain components of the graded curvature tensor, a condition that plays a role similar to the closedness of a graded symplectic form in graded symplectic geometry: It amounts to determining a graded metric by the data {g, ω, Δ′}, whereg is a metric tensor onM, ω 0 is a fibered nondegenerate skewsymmetric bilinear form on the Batchelor bundleE → M, and Δ′ is a connection onE satisfying Δ′ω = 0. Odd metrics are also studied under the same criterion and they are specified by the data {κ, Δ′}, with κ ∈ Hom (TM, E) invertible, and Δ′κ = 0. It is shown in general that even graded metrics of constant graded curvature can be supported only over a Riemannian manifold of constant curvature, and the curvature of Δ′ onE satisfiesRΔ′ (X,Y)2 = 0. It is shown that graded Ricci flat even metrics are supported over Ricci flat manifolds and the curvature of the connection Δ′ satisfies a specific set of equations. 0 Finally, graded Einstein even metrics can be supported only over Ricci flat Riemannian manifolds. Related results for graded metrics on Ω(M) are also discussed.

Fermat's principle, the general eikonal equation, and space geometry in a static anisotropic medium

Abstract: Fermat's principle and the optical metric are generalized to the case of an anisotropic medium The metric tensor of a three-dimensional Riemannian manifold is related to the dielectric tensor of the medium The general eikonal equation in a static anisotropic medium is derived The expressions for the curvature tensor and the curvature scalar that characterize the geometrical structure of a three-dimensional manifold are given For an isotropic medium the derived expressions for the curvature tensor and curvature scalar reduce to the previous results

A characterization of real hypersurfaces in complex space forms in terms of the Ricci tensor

Abstract: We study real hypersurfaces of a complex space form Mn(c), c ≠ 0 under certain conditions of the Ricci tensor on the orthogonal distribution T 0.

Obtaining holonomy from curvature

Abstract: In physical applications of differential geometry, one sometimes wishes to compute the holonomy group of a Riemannian manifold from local data, such as the curvature tensor. In general, this can be a complicated problem, but we show that, in cases of most interest in physics, the holonomy group can be obtained directly from the Lie algebras generated by the curvature tensor.

Gravity on parallelizable manifold

Abstract: General relativity postulates that the gravity field is defined on a Riemannian manifold The field equations are $R^\mu_ u = 0$ ie Ricci's curvature tensor vanishes The field equations have to be augmented by natural physical requirements like orientability, time orientability and existence of a spinorial structure Moreover, it is impossible to define the energy of the gravity field only by the metric tensor We suggest to impose an additional structure and consider parallelizable manifold ie manifolds for which a smooth field of frames exists The derivation of the field equations is by an action principle A Lagrangian, which is quadratic in the differentials, is defined The minimum of the action is achieved at a $16 \times 16$ second order quasi linear system It is of Laplacian type The system admits a unique exact solution for a centrally symmetric, static and assimptotically flat field The resulting metric is the celebrated Rosen metric, which is very close to the Schwarzchild metric The two are intrinsicly different since the scalar curvature of the Schuarzschild metric is zero, in contrast to Rosen's which is positive The suggested field admits black holes which are briefly discussed

Compact Hermitian surfaces of Einstein type with respect to the Hermitian connection

Abstract: Two Einstein-type conditions for the Hermitian curvature tensor are considered on a compact Hermitian surface: It is proved that if the symmetric part of the Ricci tensors is a scalar multiple of the metric with a negative constant, then the metric is Kaehler. If the Hermitian surface satisfies the Hermite-Einstein condition with a non positive constant, then the metric is Kaehler.

Asymptotic Curvature of Θ-Metrics

Abstract: We give an explicit and reasonably simple expression for the curvature tensor of a Θ-metric at boundary points, in terms of the metric tensor and invariants of the Θ-structure. We examine the behavior of the induced metric on level sets of a defining function near the boundary and describe the asymptotic behavior of its curvature tensor. Some applications of these results are given.

Analogy between real irreducible tensor operator and molecular two-particle operator

Abstract: . Molecular matrix elements of a physical operator are expanded in terms of polycentric matrix elements in the atomic basis by multiplying each by a geometrical factor. The number of terms in the expansion can be minimized by using molecular symmetry. We have shown that irreducible tensor operators can be used to imitate the actual physical operators. The matrix elements of irreducible tensor operators are easily computed by choosing rational irreducible tensor operators and irreducible bases. A set of geometrical factors generated from the expansion of the matrix elements of irreducible tensor operator can be transferred to the expansion of the matrix elements of the physical operator to compute the molecular matrix elements of the physical operator. Two scalar product operators are employed to simulate molecular two-particle operators. Thus two equivalent approaches to generating the geometrical factors are provided, where real irreducible tensor sets with real bases are used.

Compactness Theorems for Critical Riemannian 4-Manifolds with Integral Bounds on Curvature

Abstract: In this article, we prove the compactness of the set of critical 4-manifolds with Lp-bound on the negative part of the Ricci curvature tensor, p > 2. In earlier work we proved this under the assumption that the Ricci curvature is pointwise bounded from below.