Showing papers on "Riemann curvature tensor published in 1988"

Conformal Geometry from the Riemannian Viewpoint

Abstract: When a conformai structure on a manifold is defined by a Riemannian metric g , how to detect conformai flatness on g ? The answer, due to Weyl and Schouten, is given in § C, and some applications are derived in § D. It turns out that the three dimensional case, i.e. the case where the curvature tensor is determined by the Ricci tensor, needs a special treatment. An example of that situation is given in § E. We also give some global properties of compact conformally flat manifolds: the nullity of their Pontryagin numbers, (Chern-Simons), a vanishing theorem for middle-dimensional cohomology when the scalar curvature is positive (Bourguignon)and a structure’ theorem when the scalar curvature is zero.

Construction of manifolds of positive scalar curvature

Abstract: We prove that a regular neighborhood of a codimension > 3 subcomplex of a manifold can be chosen so that the induced metric on its boundary has positive scalar curvature. A number of useful facts concerning manifolds of positive scalar curvature follow from this construction. For example, we see that any finitely presented group can appear as the fundamental group of a compact 4-manifold with such a metric. 0. Outline of results. We give a new method for constructing complete Riemannian manifolds of positive scalar curvature and use it to continue the investigation of properties of positive scalar curvature. Our construction uses the idea that manifolds having spheres of dimension > 2 as "factors" will admit metrics of positive scalar curvature if the spheres can be made to carry sufficient positive curvature to dominate any negative curvature. Most of the known methods for constructing manifolds of positive scalar curvature employ this same idea. For example, any manifold of the form M X S2 can be given a warped-product metric of positive scalar curvature by suitably adjusting the radius of the S2-factor. Similarly, by deforming the standard metric on S3-{point} in a small neighborhood of the point and using the S2-factor to carry positive curvature around the corner we can construct a complete metric of positive scalar curvature on R3. This same idea was used by Gromov and Lawson [GL] and Schoen and Yau [SY] in proving that codimension > 3 surgeries on a manifold of positive scalar curvature yields a manifold which also carries positive scalar curvature. In this paper we generalize the above techniques to cover any manifold formed as the boundary of a regular neighborhood of a subcomplex K of a manifold M. If the codimension of K > 3 this boundary looks locally like K X S2, and so should carry positive scalar curvature. THEOREM 1. Let M be an n-dimensional Riemannian manifold with a fixed smooth cell decomposition and K a codimension q > 3 subcomplex of M. Then there is a regular neighborhood U of K in M so that the induced metric on the boundary dU has positive scalar curvature. An easy consequence of this theorem is the following. COROLLARY 2. Let 7r be a finitely presented group. Then there exists a compact 4-manifold M of positive scalar curvature with 7rl (M) = 7r. This fact is interesting since it is generally believed that manifolds that are "large" in some sense should not adrrlit metrics of positive curvature. Corollary 2 Receiv?d by the editors Septem})er 25, 1985 and, ill revised form, July 17, 1986. 1980 M(lthf'rB(lti('.s.S?l{Jjf'('t (l(l.N'.N'iJl('(ltiOn (1985 RfviSion). Primary 53C20. (r)1988 Americatl Mathematic.al Society 0002-9947/88 $1.00 + $.25 per page

On potentials for several classes of spinor and tensor fields in curved spacetimes

Abstract: Already known results with respect to the existence of a vector potential for the Maxwell field tensor and a tensor potential for Weyl's conformal curvature tensor in four-dimensional spacetimes are generalized. It is shown that there exists a spinor potential of type (n−1,1) for any totally symmetric spinor field of rankn. From this theorem we deduce a series of corollaries, for example, that every antisymmetric tensor of second rank admits a linear representation in terms of the first derivatives of two vector fields. Further, some investigations are made on the existence of potentials for arbitrary symmetric spinors of type (n, m).

A compactification of a manifold with asymptotically nonnegative curvature

Abstract: On etudie une classe de varietes de Riemann (ouverte) de courbure asymptotiquement non negative

Examples of complete manifolds of positive Ricci curvature with nilpotent isometry groups

Abstract: On the other hand, every finitely generated subgroup of the fundamental group of any complete manifold with Ric > 0 {K > 0) is nilpotent (abelian) up to finite index [6, 5, 4]. PROOF OF THE THEOREM. Our construction is inspired by [2]. We first apply an observation in [3, pp. 126-127] to obtain a family of almost flat metrics gr on L, 0 < r < oo. Choose a triangular basis {Xi,...,Xn} for the Lie algebra / of L, i.e., [X,X^] € h-i whenever X € /, and U-i is spanned by X i , . . . ,X j_ i . For X = E ? = i « < * set ||X|| == £?=i*?(r)a?, where h{(r) = (1 + r 2 ) \" \" ' , and an — a > 0, 2ai — 4c*i+i = 1, 1 < i < n — 1. The above norm gives rise to a corresponding almost flat left invariant metric gr. Then (1) iRMX^cU+r)-,

Curvature structure in general relativity

Abstract: This paper investigates the extent to which the curvature structure of space‐time determines the metric stucture. It continues the work of earlier papers by prescribing the curvature structure and the curvature covariant derivatives up to certain orders. It is shown that, with the exception of the so‐called generalized pp waves, the curvature and its first and second covariant derivatives essentially determine the metric up to coordinate transformations.

The Geometrodynamic Content of the Regge Equations as Illuminated by the Boundary of a Boundary Principle

Abstract: In this paper the principle that the boundary of a boundary is identically zero (∂○∂≡0) is applied to a skeleton geometry. It is shown that the left-hand side of the Regge equation may be interpreted geometrically as the sum of the moments of rotation associated with the faces of a polyhedral domain. Here the polyhedron, warped though it may be, is located in a lattice dual to the original skeleton manifold. This sum is related to the amount of energy-momentum (E-p) associated to the edge in question. In the establishment of this equation the ordinary Bianchi identity is rederived by applying the principle that the (∂○∂≡0) in its (1–2–3)-dimensional formulation to polyhedral domain. Steps toward the derivation of the contracted Bianchi identity using this principle in its (2–3–4)-dimensional form are discussed. Preliminary results in this direction indicate that there should be one vector identity per vertex of the skeleton geometry. “Space acts on matter, telling it how to move. In turn, matter reacts back on space, telling it how to curve.”—Ref. 3, Chap. 1.

Generalized bochner theorems and the spectrum of complete manifolds

Abstract: Bochner's theorem that a compact Riemannian manifold with positive Ricci curvature has vanishing first cohomology group has various extensions to complete noncompact manifolds with Ricci possibly negative. One still has a vanishing theorem for L2 harmonic one-forms if the infimum of the spectrum of the Laplacian on functions is greater than minus the infimum of the Ricci curvature. This result and its analogues for p-forms yield vanishing results for certain infinite volume hyperbolic manifolds. This spectral condition also imposes topological restrictions on the ends of the manifold. More refined results are obtained by taking a certain Brownian motion average of the Ricci curvature; if this average is positive, one has a vanishing theorem for the first cohomology group with compact supports on the universal cover of a compact manifold. There are corresponding results for L2 harmonic spinors on spin manifolds.

Holonomy groups in general relativity

Abstract: The infinitesimal holonomy group structure of space‐time is discussed and related to the Petrov type of the Weyl tensor and the algebraic (Segre) type of the energy‐momentum tensor. The number of covariant derivatives of the curvature tensor required to determine the infinitesimal holonomy group is determined in each case and the complete classification scheme is tabulated. Some special cases of physical interest are investigated in more detail. A geometrical approach is followed throughout.

Energy conditions in standard static spacetimes

Abstract: Let (H, h) be a Riemannian manifold and letf∶H→(0,∞) be a smooth function. The Lorentzian warped product (a,b) f ×H, -∞⩽a

Curvature of generic space‐times in general relativity

Abstract: Transversality theory is used to study the curvature of generic space‐times in general relativity and some applications to unique determination of the metric by the curvature are given.

The nonimbeddability of real hypersurfaces in spheres

Abstract: It is shown that there exist real analytic real hypersurfaces in CG which cannot be locally holomorphically imbedded in any finite dimensional sphere S2N-1 C C2N. In [1], Chern and Moser developed a theory of local invariants of real hypersurfaces in complex manifolds. In doing so, they stressed the analogies between pseudoconformal geometry and Riemannian geometry, defining analogues of the Riemannian curvature tensor, the Levi-Civita connection, geodesics, etc. These analogies are in general quite weak (e.g., "geodesics" may spiral, see [2]). The purpose of this note is to show that there is another case where the analogy breaks down. It is a classical result that any analytic metric can be locally induced by the flat metric, i.e., given any analytic metric there exists locally an isometric imbedding into some Euclidean space. The corresponding theorem in pseudoconformal geometry would state that every analytic real hypersurface could be (locally) holomorphically imbedded in the unit sphere in some Cn. The two theorems of this paper give counterexamples to this. It should be noted that if a global holomorphic imbedding of the boundary of a strictly pseudoconvex domain into a unit sphere exists, it naturally extends to a proper holomorphic mapping of the whole domain into the unit ball. Lempert [4] has shown that any smoothly bounded strictly pseudoconvex domain can be properly and holomorphically imbedded in the unit ball in infinite dimensional space (i.e., 12). The results here indicate that Lempert's theorem cannot be extended to imbeddings in finite dimensional space. THEOREM. Given integers N > n > 2, there exists a smooth, strictly pseudoconvex, real analytic (in fact, polynomial) real hypersurface M in Cn that cannot be locally analytically imbedded in the unit sphere in CN. PROOF. We shall examine the problem of mapping a real hypersurface M into CN so that the image is tangent to the sphere to order d. It will suffice to show that by taking d large enough, we can find a real hypersurface M that cannot be imbedded in CN, tangent to the unit sphere to order d. Let V be the space of real-valued polynomials F(z', z', u) of degree less than or equal to d that vanish at (0,0,0), W the space of complex polynomials f (z', w), ct = 1, ... , N, of degree less than or equal to d such that Za Ifa(0, 0)12 1 = 0. Received by the editors March 25, 1987 and, in revised form, July 6, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 32F25; Secondary 53B25, 53B15. ?1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page

The Riemann and extrinsic curvature tensors in the Regge calculus

Abstract: A detailed analysis of the Riemann tensor in the neighbourhood of one bone and of the extrinsic curvature in the neighborhood of one triangular face in a simplicial geometry is presented. Unlike most previous analyses this analysis makes no reference to any particular choice of smoothing scheme. Explicit formulae will be presented for both the Riemann and extrinsic curvature tensors. These results are applied, using the formalism developed in an earlier paper, in deriving an exact formula for the integral extrinsic curvature. It is argued that for integrals of R2, R3, . . ., contributions must be expected from the legs, vertices, . . ., rather than just from the bones.

The Geometry of the Virasoro Group for Physicists

Abstract: Diff(S 1), the group of reparametrizations of the circle, is known as the Virasoro group in string theory. Reparametrizations keeping fixed a point of the circle form the quotient space Dif f(S 1)/S 1. The geometry of this space is relevant for string theory and string field theory. We describe this space as an infinite dimensional complex manifold with a Kahler metric and compute its Riemann tensor and its Ricci tensor.

Some new cosmological results of quadratic Lagrangians.

Abstract: We present the field equations derived from a gravitational Lagrangian which includes a general combination of terms quadratic in the curvature tensor, assuming a space-time with one timelike dimension and two maximally symmetric spacelike subspaces, with arbitrary dimensions d and D. We then restrict ourselves to the case d = 3, and find a general solution for the case of a radiation-dominated universe with a static pressureless internal space. Conditions for the existence of Friedmann-type solutions are derived and discussed. Finally we study the equations, again with a static internal space, for pressureless matter in both subspaces.

Weyl curvature tensor in static spherical sources.

Abstract: The role of the Weyl curvature tensor in static sources of the Schwarzschild field is studied. It is shown that in general the contribution from the Weyl curvature tensor (the ``purely gravitational field energy'') to the mass-energy inside the body may be positive, negative, or zero. It is proved that a positive (negative) contribution from the Weyl tensor tends to increase (decrease) the effective gravitational mass, the red-shift (from a point in the sphere to infinity), as well as the gravitational force which acts on a constituent matter element of a body. It is also proved that the contribution from the Weyl tensor always is negative in sources with surface gravitational potential larger than (4/9. It is pointed out that large negative contributions from the Weyl tensor could give rise to the phenomenon of gravitational repulsion. A simple example which illustrates the results is discussed.

An improved algorithm for quartic equation classification and Petrov classification.

Abstract: The determination of multiplicities of the roots of quartic equations with (in general) nonconstant coefficients is studied in the context of the Petrov classification of the Weyl conformal curvature tensor. A history of existing algorithms for this determination is given. An alternative algorithm is described and a qualitative comparison to the above-mentioned algorithms given. Following some notes on the actual computer implementation, a quantitative comparison is made between three of the algorithms, using the symbolic computer language Maple. The algorithm is also implemented in the symbolic language MuSimp.

Weyl tensor and gravitational entropy.

Abstract: There is a conjecture due to Penrose that there may be a relation between the Weyl curvature tensor and gravitational entropy. In this paper, this conjecture is studied in the context of a specific model, the Gowdy cosmology. The square of the curvature is calculated as an operator and its expectation values in states of clumped and unclumped gravitons are calculated. The results indicate that the curvature contains information about the entropy of the gravitational field.

Cosmology in ten dimensions with the generalised gravitational Lagrangian

Abstract: A cosmological model in ten dimensions is proposed, based on a double fibre bundle V4*SU(2)*SU(2) leading to the Higgs potential and spontaneous symmetry breaking. The Lagrangian of the theory contains all the invariants of the ten-dimensional Riemann tensor up to the third, leading to the second-order differential equations. The static solutions of the system are obtained and discussed. The possibility of joining the four-dimensional Friedmann solutions from an unstable non-singular initial state through an inflationary stage is considered.

The continuous determination of spacetime geometry by the Riemann curvature tensor

Abstract: It is shown that generically the Riemann tensor of a Lorentz (or positive definite) metric on an n-dimensional manifold (n>or=4) determines the metric up to a constant factor and hence determines the associated torsion-free connection uniquely. The resulting map from Riemann tensors to connections is continuous in the Whitney Cinfinity topology (1957) but, at least for some manifolds, constant factors cannot be chosen so as to make the map from Riemann tensors to metrics continuous in that topology. The latter map is, however, continuous in the compact open Cinfinity topology so that estimates of the metric and its derivatives on a compact set can be obtained from similar estimates on the curvature and its derivatives.

Dust EPL cosmologies

Abstract: A formalism based on the description of a geometry in terms of the curvature tensor and its covariant derivatives is used to show that there are no tilted dust exact power law (EPL) cosmologies.

The isometry groups of compact manifolds with negative Ricci curvature

Abstract: We estimate the order of the isometry groups of compact manifolds with negative Ricci curvature in terms of geometric quantities: the sectional curvature, the Ricci curvature, the diameter, and the injectivity radius.

Curvature tensors unified field equations on SEX n

Abstract: We study the curvature tensors and field equations in then-dimensional SE manifold SEXn. We obtain several basic properties of the vectorsS λ andU λ and then of the SE curvature tensor and its contractions, such as a generalized Ricci identity, a generalized Bianchi identity, and two variations of the Bianchi identity satisfied by the SE Einstein tensor. Finally, a system of field equations is discussed in SEXn and one of its particular solutions is constructed and displayed.

Metrics of positive scalar curvature on spheres and the Gromov-Lawson conjecture

Abstract: Boguslaw Hajduk Institute of Mathematics, Wrociaw University, pl. Grunwaldzki 2/4, PL-50-384 Wrodaw, Poland 1. Introduction In [2] and [9] an ingenious procedure is given to construct a Riemannian metric of positive scalar curvature on a manifold obtained by surgery from one which already has such a metric. With some improvements of [1] this may be summarized as follows. 1.1. Theorem. Let (M, g) be a compact n-dimensional Riemannian manifold with positive scalar curvature and let W be a cobordism from M to M' such that W admits a handle decomposition on M with no handles of index greater than n- 2 (i.e. there exists a Morse function on W which is minimal on M and critical points have indices <-_ n- 2). Then there exists a metric of positive scalar curvature on W which extends g and is product on a collar of MUM'. This construction applied to the cobordism with one handle of index 1 between the disjoint sum MuN and the connected sum M4#N shows imme- diately that the connected sum is well defined for manifolds of positive scalar curvature. Furthermore, this operation gives an abelian group structure in the set n~ of concordance classes of positive scalar curvature metrics on S n, with the zero class represented by the standard metric gcan [1]. We say that two metrics go, gl of positive scalar curvature on M are concordant if there exists a metric g of positive scalar curvature on M x [0,1] such that glM x {i} =gi, i=0, 1, and g is product near M x d[0,1]. Our aim is to show how this group, or its subgroup ~ of classes of those metrics which are boundary restrictions of metrics of positive scalar curvature on compact spin manifolds, is related to some questions concerning positive scalar curvature. IfX n is a 2-connected dosed manifold, B a smooth n-baU in X, then by Morse- Srnale theory and Theorem 1.1 there is a metric of positive scalar curvature on X- Int B which is product near S n- i = O(X - Int B). This metric induces a metric of positive scalar curvatures on S "- 1. Our basic observation is that the concordance class 6(X) of the induced metric depends only on the spin cobordism class of X and 6(X)--. 0 if and only if any l-connected manifold spin cobordant to X admits a

4-planar geodesic Kaehler immersions into a complex projective space

Abstract: If f is a proper 4-planar geodesic Kaehler immersion of a connected complete Kaehler manifold MI (n > 2) into CPm(c), then Mn = CP1(c/4) and f is equivalent to the 4th Veronese map. 0. Introduction. Let M be a Riemannian manifold. A curve r: I -* M defined on an open interval I is said to be d-planar if there exist an open interval I, (s E I, C I) and a d-dimensional totally geodesic submanifold P8 for each s E I such that r(I) C P. Moreover, a d-planar curve r is said to be proper if it is not (d 1)-planar on each open subinterval of I. An isometric immersion f: M -* M of a Riemannian manifold M is called a (resp. proper) d-planar geodesic immersion if r = f o ais (resp. proper) d-planar geodesic for every geodesic 'y: I -* M. 1-planar geodesic immersions are totally geodesic. 2-planar geodesic immersions into real space forms were classified in [7] (for other treatment, see [1]). When the ambient manifold is a complex projective space COPm (c) with constant holomorphic sectional curvature c, 2-planar and odd order proper planar geodesic Kaehler immersions were classified in [5 and 6], respectively. In this paper, we shall study proper 4-planar geodesic Kaehler immersions into CP' (c). 1. Notation and basic equations (cf. [2]). For a Kaehler immersion f: M CP' (c), the second fundamental form and Weingarten map corresponding to a normal vector field ( will be denoted by H and A~, respectively. Gauss and Weingarten's equations are given by (1.1) vxY = VXY + H(X, Y), Vx, = -A~X + Vx for all tangent vector fields X and Y on M, where V, V, and VI denote the covariant differentiation of M, M, and the normal bundle, respectively. Let R be the curvature tensor and J the complex structure of M. The structure equation of Gauss is given by (1.2) R(X, Y)Z (c/4){(Y, Z)X (X, Z)Y + (JY, Z)JX (JX, Z)JY 2(JX, Y)JZ} + AH(Y,Z)X AH(X,Z)Y. Received by the editors January 20, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 53C40.