Showing papers on "Riemann curvature tensor published in 1979"
TL;DR: Schoen and Yau as mentioned in this paper showed that any complete conformally flat manifold with non-negative scalar curvature is the quotient of a domain in Sn by a discrete subgroup of the conformal group.
Abstract: Publisher Summary This chapter discusses some recent results by Richard Schoen and Shing-Tung Yau on the structure of manifolds with positive scalar curvature. The chapter presents theorems which are felt to provide a more complete picture of manifolds with positive scalar curvature: (1) let M be a compact four-dimensional manifold with positive scalar curvature. Then there exists no continuous map with non-zero degree onto a compact K(π,1). (2) Let M be n-dimensional complete manifold with non-negative scalar curvature. Then any conformed immersion of M into Sn is one to one. In particular, any complete conformally flat manifold with non-negative scalar curvature is the quotient of a domain in Sn by a discrete subgroup of the conformal group. (3.) Let M be a compact manifold whose fundamental group is not of exponential growth. Then unless M is covered by Sn, Sn–1 x S1 or the torus, M admits no conformally flat structure.
627 citations
156 citations
TL;DR: In this article, the authors studied the properties of harmonic maps from a compact manifold M into a manifold N with non-positive curvature and showed that the space is a compact connected totally geodesic submanifold of N and is a point if r,(N) has no non-trivial abelian subgroup and the image of M is not a point or a circle.
142 citations
TL;DR: In this article, the authors studied the relationship of A to the geometry of X and determined the asymptotic growth of A as a subset of A(S) under the assumption that X is compact and defined the spectrum A of X as the set of those elements of A (S) for which one can find a nonzero eigenfunction defined on X.
Abstract: Let S be a Riemannian symmetric space of noncompact type, and let G be the group of motions of S. Then the algebra L-~ of G-invariant differential operators on S is commutative, and its spectrum A(S) can be canonically identified with ~/w where ~ is a complex vector space with dimension equal to the rank of S, and to is a finite subgroup of G L ( ~ ) generated by reflexions. Let P be a discrete subgroup of G that acts freely on S and let X = E \\ S . Then the members of 5~ may be regarded as differential operators on X. Let us now assume that X is compact and define the spectrum A of X as the set of those elements of A(S) for which one can find a nonzero eigenfunction defined on X. In this paper we study the relationship of A to the geometry of X and determine the asymptotic growth of A as a subset of A(S). In subsequent papers we plan to study the asymptotic behaviour of the eigenfunctions and to examine the problem of obtaining improvements on the error estimates. It is well-known that G, which is transitive on S, is a connected real semisimple Lie group with trivial center, and that the stabilizers in G of the points of S are the maximal compact subgroups of G. So we can take S = G/K, X =F\\G/K, where K is a fixed maximal compact subgroup of G, and F is a discrete subgroup of G containing no elliptic elements (= elements conjugate to an element of K) other than e, such that F\\G is compact. Let G = K A N be an Iwasawa decomposit ion of G; let o be the Lie algebra of A; and let to be the Weyl group of (G, A). If we take ,~to be the dual of the complexification a c of a, then A ( S ) ~ / w canonically. In what follows we shall commit an abuse of notation and identify A(S) with ,~, but with the proviso that points of ~ in the same w-orbit represent the same element of A(S).
130 citations
TL;DR: The Dirichlet sub-solution for nonnegative sectional, Ricei, and bisectional curvature problems was studied in this paper, where the main point of this procedure is to sidestep arguments involving continuous functions by working with differentiable functions alone.
Abstract: A standard technique in classical analysis for the study of eontinous sub-solutions of the Dirichlet problem for second order operators may be illustrated as follows. Suppose it is to be shown that a continuous real function ](x) is convex (respectively, striely convex) at x0; then it suffices to produce a C ~ function g(x) such that g(x)<<.](x) near x 0 and g(Xo) =/(x0), and such that 9\"(xo) >/0 (respectively g\"(xo) >1 some fixed positive constant). The main point of this procedure is to sidestep arguments involving continuous functions by working with differentiable functions alone. Now in global differential geometry, the functions that naturally arise are often continuous but not differentiable. Since much of geometric analysis reduces to second order elliptic problems, this technique then recommends itself as a natural tool for overcoming this difficulty with the lack of differentiability. In a limited way, this technique has indeed appeared in several papers in complex geometry (e.g. Ahlfors [1], Takeuchi [20], Elenewajg [7] and Greene-Wu [11]; cf. also Suzuki [19]). The main purpose of this paper is to broaden and deepen the scope of this method by making it the central point of a general study of nonnegative sectional, Ricei or bisectional curvature. The following are the principal theorems; the relevant definitions can be found in Section 1. Let M be a noncompact complete Riemannian manifold and let 0 E M be fixed. Let {Ct}tG1 be a family of closed subsets of M indexed by a subset I of R. Assume that et = d(0, C t ) ~ as t ~ , where d(p, q) will always denote the distance between p, qEM relative to the Riemannian metric. The family of functions ~t: M-~R defined by ~t(P)=
104 citations
61 citations
47 citations
CERN1
TL;DR: In this article, the supersymmetric extension of invariants quadratic in the Riemann curvature tensor for SO(N )-extended supergravity was studied.
43 citations
TL;DR: In this paper, the mutual inequivalence of g-variation and P-variations of a given action S is discussed, where the components of linear connection Gamma mkl being required to be symmetric.
Abstract: The author deals with the mutual inequivalence of g-variation and P-variation of a given action S, the components of linear connection Gamma mkl being required to be symmetric. Under g-variation S is required to be stationary with respect to variations of the metric tensor gij, the Gamma mkl being taken to be Christoffel symbols from the outset, whereas under P-variations the gij and Gamma mkl are initially regarded as mutually independent of S is required to be stationary with respect to independent variations of these quantities. The discussion is illustrated at length by examples in which the Lagrangian of S is one or another of a set of homogeneous or inhomogeneous quadratic invariants of the Riemann tensor.
40 citations
40 citations
TL;DR: In this paper, the Lagrangian is generated by a variational principle, which is solely a function of curvature quantities and the metric tensor gij of an A4*, i.e. of a linearly connected space A4 with symmetric connection on which an additional symmetric tensor field gij is defined.
Abstract: The Einstein-Proca field is 'geometrised' in the sense that the joint field equations are generated by a variational principle, the Lagrangian of which is solely a certain function of curvature quantities and the metric tensor gij of an A4*, i.e. of a linearly connected space A4 with symmetric connection on which an additional symmetric tensor field gij is defined.
TL;DR: In this article, it has been shown that in general the parameter space is not necessarily of a constant curvature and that, if the correlation coefficient vanishes, the parameter spaces becomes an Einstein space.
TL;DR: In this paper, the curvature tensor is expressed as a function of the motion of the source of this curvature, and the behavior of the orbit of two particles weakly interacting gravitationally, with radiation reaction taken into account, is used to compute the asymptotic behavior of corresponding curvatures along past-directed null straight lines.
Abstract: The standard weak-field, slow-motion approximation to Einstein's relativistic theory of gravitation is used to express the curvature tensor, up to order ${r}^{\ensuremath{-}5}$ on a flat background space-time, as a functional of the motion of the source of this curvature. The behavior, in the distant past, of the orbit of two particles weakly interacting gravitationally, with radiation reaction taken into account, is then used to compute the asymptotic behavior of the corresponding curvature tensor along past-directed null straight lines in the flat background. It is found, on the one hand, that the falloff of the curvature is fast enough to guarantee satisfaction of a condition to exclude incoming gravitational radiation. On the other hand, the falloff is slower than would have been expected if the conformally rescaled curvature tensor had been regular on the hypersurface at past null infinity of the flat background.
TL;DR: In this paper, an algebraic classification of the Ricci tensor is given in terms of its invariant two-space structure, which involves classifying a complex fourth-order tensor which is algebraically equivalent to the trace-free Ricci Tensor and which has all the algebraic symmetries of the (complex) Riemann tensor.
Abstract: An algebraic classification of the Ricci tensor is given in terms of its invariant two-space structure. The method involves classifying a complex fourth-order tensor which is algebraically equivalent to the trace-free Ricci tensor and which has all the algebraic symmetries of the (complex) Riemann tensor.
TL;DR: In this article, it was shown that Cotton's tensor density is not the Euler-Lagrange expression corresponding to a scalar density built from one metric tensor.
Abstract: It is well known that a necessary and sufficient condition for the conformal flatness of a three‐dimensional pseudo‐Riemannian manifold can be expressed in terms of the vanishing of a third‐order tensor density concomitant of the metric which has contravariant valence 2. This was first discovered by Cotton in 1899. It is shown that Cotton’s tensor density is not the Euler–Lagrange expression corresponding to a scalar density built from one metric tensor. This tensor density is shown to be uniquely characterized by its conformal properties coupled with the demand that it be differentiable for arbitrary metrics.
TL;DR: In this article, the topological and geometric relations of the energy-momentum tensor of zero-point vacuum vibrations of a scalar field in two-dimensional manifolds were investigated.
Abstract: We consider the topological and geometric relations of the energy-momentum tensor of zero-point vacuum vibrations of a scalar field in two-dimensional manifolds. It is shown that the energy density can change sign under continuous deformation of the region.
TL;DR: In this article, the curvature operator of a Riemannian manifold is investigated using the exterior algebra of a vector space, and the results about the Euler characteristic of a compact manifold are derived.
Abstract: Properties of the exterior algebra of a vector space are used to investigate the curvature operator of a Riemannian manifold. Induced inner products and linear maps are used to establish results about the Euler characteristic of a compact manifold. An open problem about the decomposition of operators on A 2 V is discussed. This problem arises in the study of the codimension needed for isometric embeddings. A new algebraic consequence of the first Bianchi identities is established.
TL;DR: In this article, the connection between the critical point structure of the Riemannian curvature function and the Petrov classification of the Ricci tensor has been investigated, and a similar function is defined whose critical point structures are connected with the algebraic classification of RicCI tensors.
Abstract: Some theorems proved by Thorpe concerning the connection between the critical point structure of the Riemannian (sectional) curvature function and the Petrov classification are extended. A similar function is defined whose critical point structure is connected with the algebraic classification of the Ricci tensor.
TL;DR: In this article, the analytic dependences of the vacuum averages of the energy-momentum tensor on geometry and topology are found for scalar, electromagnetic, and massless gluon fields in bounded three-dimensional manifolds.
Abstract: The analytic dependences of the vacuum averages of the energy-momentum tensor on geometry and topology are found for scalar, electromagnetic, and massless gluon fields in bounded three-dimensional manifolds. Application of the results obtained to the “bag” model in the theory of hadrons is discussed.
TL;DR: In this paper, the authors describe a straightforward approach to study the motion of the sources of some Robinson-Trautman gravitational fields (1962) in linearised gravity, and show that the solution agrees with the exact solution of Levi-Civita to this problem, in the linear approximation.
Abstract: The authors describe a straightforward approach to studying the motion of the sources of some Robinson-Trautman gravitational fields (1962) in linearised gravity. It involves expanding the Robinson-Trautman line-element about Minkowskian space-time in powers of a small parameter (the 'mass' of the source). They solve the linearised field equation in vacuo by first specifying the source world-line in the background Minkowskian space-time. Functions of integration are determined by the requirement that terms be excluded from the field (Riemann tensor) of the particle which are singular along null-rays emanating into the future from events on the source world-line in the background space-time. As an example they take the world-line to be the history of a uniformly accelerated particle. They show that the solution agrees with the exact solution of Levi-Civita to this problem, in the linear approximation.
TL;DR: In this paper, the behavior of the electromagnetic field tensor is examined under symmetry mappings, including Ricci collineations, curvature and Ricci curvature, and the Ricci Ricci Collineations family.
Abstract: General properties of Einstein-Maxwell spaces, with both null and nonnull source-free Maxwell fields, are examined when these space-times admit various kinds of symmetry mappings. These include Killing, homothetic and conformal vector fields, curvature and Ricci collineations, and mappings belonging to the family of contracted Ricci collineations. In particular, the behavior of the electromagnetic field tensor is examined under these symmetry mappings. Examples are given of such space-times which admit proper curvature and proper Ricci collineations. Examples are also given of such space-times in which the metric tensor admits homothetic and other motions, but in which the corresponding Lie derivatives of the electromagnetic Maxwell tensor are not just proportional to the Maxwell tensor.
TL;DR: In this paper, the structure of the Riemannian curvature tensor is investigated and interpreted in terms of the infinitesimal holonomy groups (IHG).
Abstract: This paper further investigates and interprets the structure of Einstein-Maxwell space-times in terms of the infinitesimal holonomy groups (IHG) of theC
∞ Riemannian connection. In particular, this investigation provides a fundamental physical classification of Einstein-Maxwell space-times in terms of the IHG group structure of the Riemann curvature tensor. It will be shown that the Maxwell fieldsF
μλ and *F
μλ of a given Einstein-Maxwell space-time define a representation of a subalgebra of the Lie algebra of the IHG. The main results of this paper are stated in the form of two theorems and a corollary. Also the results obtained here indicate that physical insight could be gained by studying more general gauge fields in terms of the IHG of the relevant Einstein-Yang-Mills space-times.
TL;DR: In this paper, the problem of locally imbedding a null hypersurface in a Riemannian manifold was studied and the generalized Gauss-Codazzi equations were derived.
Abstract: This paper is concerned with the problem of locally imbedding a null hypersurface in a Riemannian manifold. More precisely, on a one‐parameter family of null hypersurfaces, rigged by an arbitrary null vector field, in a four‐dimensional space–time manifold, a particular symmetric affine connection is used to derive the corresponding generalized Gauss–Codazzi equations. In addition, expressions are obtained for the projections of the Ricci tensor, which are relevant to the characteristic initial‐value problem of general relativity.
TL;DR: In this paper, a line element associated with the gravitational field of an arbitrarily accelerating body endowed with rotation was presented, and it was shown that if the body is axially symmetric, rotates uniformly about its symmetry axis, and moves along its symmetric axis when viewed in a background flat space-time and of the linearized field (Riemann tensor) of the body was free of wire singularities then the body must move with uniform acceleration.
TL;DR: In this article, it was shown that a square invariant Weyl conformal curvature tensor can lead to a Lagrangian in a variational principle for a gravitational equation in vacuum of the Bianchi identity type which is compatible with the Einstein equation.
Abstract: It is shown that a square invariant of the Weyl conformal curvature tensor can lead to a Lagrangian in a variational principle for a gravitational equation in vacuum of the Bianchi identity type which is compatible with the Einstein equation. Moreover we show that such a Lagrangian implicitly includes a conformally invariant theory characterized by two gauge fields and the metric tensor.
TL;DR: In this article, it was shown that a 3-dimensional Riemannian manifold admits locally symmetric vector fields (LSVF) and is a V(k)-space.
Abstract: It is shown that if ann-dimensional (n≧3) Riemannian manifold admitsr≧2 locally symmetric vector fields (LSVF's), then it is aV(k)-space. In particular, ifr=n−1 then the manifold is a space of constant curvature. In the case of a 3-dimensional Riemannian manifold a close connection between LSVF's and eigenvectors of the Ricci tensor is found.