Showing papers on "Riemann curvature tensor published in 1996"

M-Theory on Eight-Manifolds

Abstract: We show that in certain compactifications of ${\cal M}$-theory on eight-manifolds to three-dimensional Minkowski space-time the four-form field strength can have a non-vanishing expectation value, while an $N=2$ supersymmetry is preserved. For these compactifications a warp factor for the metric has to be taken into account. This warp factor is non-trivial in three space-time dimensions due to Chern-Simons corrections to the fivebrane Bianchi identity. While the original metric on the internal space is not K\"ahler, it can be conformally transformed to a metric that is K\"ahler and Ricci flat, so that the internal manifold has $SU(4)$ holonomy.

Level set approach to mean curvature flow in arbitrary codimension

Abstract: We develop a level set theory for the mean curvature evolution of surfaces with arbitrary co-dimension, thus generalizing the previous work [8, 15] on hypersurfaces. The main idea is to surround the evolving surface of co-dimension k in R by a family of hypersurfaces (the level sets of a function) evolving with normal velocity equal to the sum of the (d — k) smallest principal curvatures. The existence and the uniqueness of a weak (level-set) solution, is easily established using mainly the results of [8] and the theory of viscosity solutions for second order nonlinear parabolic equations. The level set solutions coincide with the classical solutions whenever the latter exist. The proof of this connection uses a careful analysis of the squared distance from the surfaces. It is also shown that varifold solutions constructed by Brakke [7] are included in the level-set solutions. The idea of surrounding the evolving surface by a family of hypersurfaces with a certain property is related to the barriers of De Giorgi. An introduction to the theory of barriers and his connection to the level set solutions is also provided.

Hyperbolic reductions for Einstein's equations

Abstract: We consider the problem of reducing initial value problems for Einstein's field equations to initial value problems for hyperbolic systems, a problem of importance for numerical as well as analytical investigations of gravitational fields. The main steps and the most important objectives in designing hyperbolic reductions are discussed. Various reductions which have already been studied in the literature or which can easily be derived from previous discussions of the field equations are pointed out and some of their specific features are indicated. We propose new reductions based on the use of the Bianchi equation for the conformal Weyl tensor. These reductions involve symmetric hyperbolic systems of propagation equations and allow a number of different gauge conditions. They use unknowns in a most economic way, supplying direct and non-redundant information about the geometry of the time slicing and the four-dimensional spacetime. Some of this information is directly related to concepts of gravitational radiation. All these reductions can be extended to include the conformal field equations. Those which are based on the ADM representation of the metric can be rewritten in flux conserving form.

On the surfactant mass balance at a deforming fluid interface

Abstract: The amount of surfactants ~surface active agents! adsorbed onto a fluid interface affects its surface tension. Thus the distribution of surfactants must be determined to find the jump in the normal and tangential stresses across the interface. Scriven ~see also Aris, Slattery, and Edwards et al.! uses differential geometry to derive the correct surface balance equation for an arbitrary surface coordinate system. Also invoking differential geometry, Waxman develops a correct form in ~‘‘fixed’’! surface coordinates that advance only normal to the surface. To arrive at this balance without appealing to differential geometry, Stone presents a simple physical derivation which leads to a form of the mass balance which is easy to solve numerically. Unfortunately, Stone’s derivation leaves the nature of the unsteady time derivative ambiguous. Here we follow the spirit set in Stone to derive geometrically the surface balance in a way that keeps the nature of the time derivative explicit. We verify that in Stone’s form the time derivative must hold the fixed coordinates constant, as the numerical implementation of this form of the mass balance actually do. We also derive a new form valid in an arbitrary surface coordinate system. Consider a fixed point A on a fluid surface with local normal n as in Fig. 1. We locate any two perpendicular planes which intersect along n. The intersection of each of these planes with the surface near the point A define curves whose unit tangents are t1 and t2 . By construction ]t1/]s152~1/R1!n and ]t2/]s252~1/R2!n, where ds1 and ds2 are differential arcs and R1~.0! and R2~.0! are the radii of curvature of the curves. Geometrically, these differential arcs are ds15R1df1 and ds25R2df2 , where df1 and df2 are the differential angles in the figure, and ]t1/]f152n and ]t2/]f252n. Thus in this locally orthogonal system, the components of the surface metric tensor aab are: Aa115R1 , a1250, and Aa225R2 and the diagonal elements simply act as scale factors. These arcs define a patch of area dA5Aa11Aa22df1df25Aadf1df2 where a is the determinant of the metric tensor. The diagonal components of the curvature tensor bab are defined by @]ta /]fa#–n 5 baa /Aaaa ~no sum on a!; so b1152R1 and b2252R2 . The curvatures are negative because as drawn in Fig. 1 both arcs are concave down with respect to the normal. If U is the instantaneous material velocity vector at the fixed point, its components along $n,t1 ,t2% are U5Us(1)t11Us(2)t21Wn, where W is the normal component and Us(1) and Us(2) are the physical components tangent to the surface. The fixed point advances along the normal ~n! as shown in the Fig. 1 a distance WDt so that the patch perimeters have lengths (R11WDt)df1 and (R21WDt)df2 at the time t1Dt; thus the change in area of the patch is WDt(R11R2)df1df2 and the per unit area per unit time rate of change is

Conformally dressed black hole in 2 + 1 dimensions.

Abstract: A three-dimensional black hole solution of Einstein equations with a negative cosmological constant coupled to a conformal scalar field is given. The solution is static, circularly symmetric, asymptotically anti-de Sitter-type and nonperturbative in the conformal field. The curvature tensor is singular at the origin while the scalar field is regular everywhere. The condition that the Euclidean geometry be regular at the horizon fixes the temperature to be $T=\frac{9{r}_{+}}{16\ensuremath{\pi}{l}^{2}}$. Using the Hamiltonian formulation including boundary terms of the Euclidean action, the entropy is found to be $\frac{2}{3}$ of the standard value ($\frac{1}{4}A$), and in agreement with the first law of thermodynamics.

Nontrivial vacua in higher-derivative gravitation.

Abstract: A discussion of an extended class of higher-derivative classical theories of gravity is presented. A procedure is given for exhibiting the new propagating degrees of freedom, at the full nonlinear level, by transforming the higher-derivative action to a canonical second-order form. For general fourth-order theories, described by actions which are general functions of the scalar curvature, the Ricci tensor and the full Riemann tensor, it is shown that the higher-derivative theories may have multiple stable vacua. The vacua are shown to be, in general, nontrivial, corresponding to de Sitter or anti-de Sitter solutions of the original theory. It is also shown that around any vacuum the elementary excitations remain the massless graviton, a massive scalar field, and a massive ghostlike spin-two field. The analysis is extended to actions which are arbitrary functions of terms of the form ${\ensuremath{ abla}}^{2k}R$, and it is shown that such theories also have a nontrivial vacuum structure.

Large manifolds with positive Ricci curvature

Abstract: The main purpose of this paper is to show that an n-dimensional manifold with Ricci curvature greater or equal to (n−1) which is close (in the Gromov– Hausdor topology) to the unit n-sphere has volume close to that of the sphere. This shows the converse of the theorem in [C1]. Namely together with [C1] it shows that an n-manifold with Ricci curvature greater or equal to (n − 1) is close to the sphere if and only if the volume is close to that of the sphere. In particular, by [P], such a manifold is homeomorphic to a sphere. Further, as an application of this and the result of [C1], we prove a Radius Theorem saying that if an n-manifold with Ricci curvature greater or equal to (n − 1) has radius almost equal to ; then the volume is close to that of the sphere. In order to obtain these results we further develop and apply the estimates of [C1]. Whereas the main concern in [C1] were with the large scale geometry the main concern of this paper is with the small scale geometry. Let !n be the volume of the round n-sphere, Sn; with sectional curvature one.

Generalized functions and distributional curvature of cosmic strings

Abstract: A new method is presented for assigning distributional curvature, in an invariant manner, to a spacetime of low differentiability, using the techniques of Colombeau's `new generalized functions'. The method is applied to show that the scalar curvature density of a cone is equivalent to a delta function. The same is true under small enough perturbations.

A note on the Gauss-Bonnet theorem for Finsler spaces

Abstract: Exactly fifty years ago, one of us gave a proof of the Gauss-Bonnet formula for Riemannian manifolds by the method of transgression ([Chl], [Ch2]), and introduced a 'total curvature' H whose properties have yet to be fully exploited. Other proofs have since been given, including one as the simplest case of the Atiyah-Singer index theorem. The Finsler side of the story is what concerns us in the following pages, and it begins with a work of Lichnerowicz's [L] in 1948. In that paper, using the Cartan connection, Lichnerowicz established a Gauss-Bonnet theorem for all Finsler surfaces [modulo the issue of Vol(x) discussed below] and also for all Cartan-Berwald spaces of even dimension greater than two. His proof was modelled after the intrinsic method just mentioned. There are several interesting issues raised by Lichnerowicz's paper. One concerns the volume Vol(x) of the unit Finsler sphere IxM in each tangent space TxM. Various attempts to understand why he assigned these volumes the constant Euclidean values (as in Riemannian geometry) have led to some developments which play a key role in our treatment here. There is also the issue which revolves around the choice of a connection. It appears that his restriction to Cartan-Berwald spaces was dictated by the structure of the curvature tensor of the Cartan connection. We will show that the use of a connection introduced by one of us [Ch3] in 1948 effects the extension of Lichnerowicz's result to a much larger class of Finsler manifolds. Finally, there is the question of where the Gauss-Bonnet integrand should live. In Riemannian geometry, it is a top degree form on the underlying manifold M. It was Lichnerowicz who proposed that for the Finsler case, little is lost by allowing this integrand to live on the projective sphere bundle SM, as

Combinatorial scalar curvature and rigidity of ball packings

Abstract: Let M be a triangulated three-dimensional manifold. In this paper we define a combinatorial analogue of scalar curvature for M, and also a combinatorial analogue of conformal deformation of the metric. We further define a functional S on the combinatorial conformal deformation space, show that S is concave, and show that critical points of S correspond precisely to metrics of constant combinatorial scalar curvature on M. These results are then applied to showing rigidity of ball packings with prescribed combinatorics (the concepts are quite similar to Colin de Verdière’s work on circle packing of surfaces [2]. See also [5] for a related variational argument). The plan of the paper is as follows. In section 1 we define the class of conformal simplices in E, and prove the necessary local versions of our results. In section 3 we extend these techniques to conformal simplices in

Stable complete surfaces with constant mean curvature

Abstract: Letx:M2→N3 be a stable immersion with constant mean curvatureH of a complete orientable surfaceM2 into a complete oriented three dimensional Riemannian manifoldN3. In this paper we prove that, ifM2 is compact andH2> −1/2 infM RiccN, thenM2 has genusg≤3, here RiccN is the Ricci curvature ofN3. We also prove that, ifM2 is complete non compact andN2 has bounded geometry, the area ofM2 is infinite in the metric induced byx. In this case, ifH2≥−1/2 infM RiccN thenx is umbilic and the equality holds.

Smoothing Riemannian Metrics with Ricci Curvature Bounds

Abstract: We prove that Riemannian metrics with an absolute Ricci curvature bound and a conjugate radius bound can be smoothed to having a sectional curvature bound. Using this we derive a number of results about structures of manifolds with Ricci curvature bounds.

Optimal Sobolev inequalities on complete Riemannian manifolds with Ricci curvature bounded below and positive injectivity radius

Abstract: The concept of best constants for Sobolev embeddings appeared to be crucial for solving limiting cases of some partial differential equations. A striking example where it has played a major role is given by the very famous Yamabe problem. While the situation is well understood for compact manifolds, things are less clear when dealing with complete manifolds. Aubin proved in '76 that optimal Sobolev inequalities are valid for complete manifolds with bounded sectional curvature and positive injectivity radius. We prove here that the result still holds if the bound on the sectional curvature is replaced by a lower bound on the Ricci curvature (a much weaker assumption). We also get estimates for the remaining constants.

Multidimensional gravity with einstein internal spaces

Abstract: A multidimensional gravitational model on the manifold M = M0 × Q n=1 Mi, where Mi are Einstein spaces (i ≥ 1), is studied. For N0 = dimM0 > 2 themodel representation is considered and it is shown that the corresponding Euclidean Toda-like system does not satisfy the Adler-van-Moerbeke criterion. For M0 = R N0 , N0 = 3, 4, 6 (and the total dimension D = dimM = 11, 10, 11, respectively) nonsingular spherically symmetric solutions to vacuum Einstein equations are obtained and their generalizations to arbitrary signatures are considered. It is proved that for a non-Euclidean signature the Riemann tensor squared of the solutions diverges on certain hypersurfaces in R N0 .

Extensions and applications of 1 + 3 decomposition methods in general relativistic cosmological modelling

Abstract: 1 + 3 “threading” decomposition methods of the pseudo-Riemannian spacetime manifold (M;g) and all itsgeometrical objects and dynamical relations with respect to an invariantly defined preferred timelike referencecongruence u=chave been useful tools in general relativistic cosmological modelling for more than three decades.In this thesis extensions of the 1+3 decomposition formalism are developed, partially in fully covariant form, andpartially on the basis of choice of an arbitrary Minkowskian orthonormal reference frame, the timelike directionof which is aligned with u=c. After introductory remarks, in Chapter 2 first an exposition is given of the general1 + 3 covariant dynamical equations for the fluid matter and Weyl curvature variables, which arise from the Ricciand second Bianchi identities for the Riemann curvature tensor of (M;g;u=c). New evolution equations arethen derived for all spatial derivative terms of geometrical quantities orthogonal to u=c. The latter are used todemonstrate in 1 + 3 covariant terms that the spatial constraints restricting relativistic barotropic perfect fluidspacetime geometries are preserved along the integral curves of u=c. The integrability of a number of differentspecial subcases of interest can easily be derived from this general result.In Chapter 3, 1 + 3 covariant representations of two classes of well-known cosmological models with abarotropic perfect fluid matter source are obtained. These are the families of the locally rotationally symmetric(LRS) and the orthogonally spatially homogeneous (OSH) spacetime geometries, respectively. Subcases arisingfrom either dynamical restrictions or the existence of higher symmetries are systematically discussed. For exam-ple, models of purely “magnetic” Weyl curvature and, in the LRS case, a transparent treatment of tilted spatialhomogeneity can be obtained. The 1 + 3 covariant discussion of the OSH models requires completion.Chapter 4 reviews the complementary 1 + 3 orthonormal frame (ONF) approach and extends it to includethe second Bianchi identities, which provide dynamical relations for the physically interesting Weyl curvaturevariables. Then, possible choices of local coordinates within the 1+3 ONF framework are introduced, taking boththe 1 + 3 threading and the ADM 3 + 1 slicing perspectives.The 1 + 3 ONF method is employed in Chapter 5 to investigate the integrability of the dynamical equationsdescribing “silent” irrotational dust spacetime geometries, for which the “magnetic” part of the Weyl curvature isrequired to vanish. Evidence is obtained that these equations may not be consistent in the generic case, but thatonly either algebraically special or spatially homogeneous classes of solutions may be covered. Furthermore, thischapter uses the extended 1 + 3 ONF dynamical equations to describe LRS models with an imperfect fluid mattersource and contrasts the perfect fluid subcase with the results obtained in Chapter 3.In Chapter 6, a brief detour is taken into considering those classical theories of gravitation in which the La-grangean density of the gravitational field is assumed to be proportional to a general differentiable function f(R)in the Ricci curvature scalar. The generalisations of the relativistic 1 + 3 covariant dynamical equations to thef(R) case are derived and a few examples of applications are commented on.Finally, Chapter 7 investigates in detail features of the dynamical evolution of the cosmological density pa-rameter in anisotropic inflationary models of Bianchi Type–I and Type–V and points out important qualitativechanges as compared to the idealised standard FLRW situation. A related analysis employing the same spacetimegeometries addresses the occurrence of restrictions on the permissible functional form of the inflationary expansionlength scale parameter Sas a consequence of the so-called reality condition for Einstein–Scalar-Field configura-tions. Again, the effect of the (exact) anisotropic perturbations on the FLRW case is thoroughly studied and foundto have significant effects. Both cases can be treated as examples of structural instability.This thesis ends with concluding remarks and an appendix section containing the conventions employed andmathematical relations relevant to derivations given in various chapters.

Comparison theorems for curves of bounded geodesic curvature in metric spaces of curvature bounded above

Abstract: Comparison and rigidity theorems are proved for curves of bounded geodesic curvature in singular spaes of curvature bounded above. Most of these estimates do not appear in the literature even for smooth curves in Riemannian manifolds. Geodesic curvature (which agrees with the usual one in the smooth case) is defined by comparison to curves of constant curvature in a model space. Two methods of comparison are used, preserving either sidelengths of inscribed triangles or arclength and chordlength. Using a majorization theorem of Reshetnyak, we obtain best possible global comparisons for arclength, chordlength, width and base angles in a CAT(K) space. A criterion for a metric ball to be a CAT(K) space is also given, in terms of the radius and the radial uniqueness property.

Einstein’s equations in the presence of signature change

Abstract: We discuss Einstein’s field equations in the presence of signature change using variational methods, obtaining a generalization of the Lanczos equation relating the distributional term in the stress tensor to the discontinuity of the extrinsic curvature. In particular, there is no distributional term in the stress tensor, and hence no surface layer, precisely when the extrinsic curvature is continuous, in agreement with the standard result for constant signature.

Mean Curvature of Riemannian Foliations

Abstract: It is shown that a suitable conformai change of the metric in the leaf direction of a transversally oriented Riemannian foliation on a closed manifold will make the basic component of the mean curvature harmonic. As a corollary, we deduce vanishing and finiteness theorems for Riemannian foliations without assuming the harmonicity of the basic mean curvature.

Combinatorics of curvature, and the bianchi identity

Abstract: We analyze the Bianchi Identity as an instance of a basic fact of com- binatorial groupoid theory, related to the Homotopy Addition Lemma. Here it becomes formulated in terms of 2-forms with values in the gauge group bundle of a groupoid, and leads in particular to the (Chern-Weil) construction of characteristic classes. The method is that of synthetic differential geometry, using "the first neighbourhood of the diagonal" of a manifold as its basic combinatorial structure. We introduce as a tool a new and simple description of wedge (= exterior) products of differential forms in this context.

Finding solutions to Einstein's equations in terms of invariant objects

Abstract: The method for constructing geometries in terms of a set consisting of the curvature tensor and a finite number of its covariant derivatives is extended to determine the isometry group and to find the full line-element. Comparisons are made with other tetrad methods. Two perfect fluid examples with LRS are given to illustrate the formalism: one static, conformally flat case and one stationary, rotating case with vanishing magnetic part of the Weyl tensor.

Hadamard-Frankel type theorems for manifolds with partially positive curvature.

Abstract: In this paper we prove some theorems that two minimal submanifolds satisfying a condition for the dimensions of the submanifolds in a Riemannian manifolds with partially positive curvature or a Kaehler manifold with partially positive holomorphic sectional curvature must intersect. Our results show that the famous Frankel theorem about intersections of minimal submanifolds in a manifold with positive curvature is generalized to the very wide class of manifolds with partially positive curvature.