Showing papers on "Riemann curvature tensor published in 1976"

Homogeneous Manifolds With Negative Curvature

Abstract: This paper solves the problem of determining which Lie groups act simply transitively on a Riemannian manifold with negative curvature. The results obtained extend those of Heintze for the case of strictly negative curvature. Using results of Wolf and Heintze, it is established that every connected, simply connected, homogeneous manifold M with negative curvature admits a Lie group S acting simply transitively by isometries and every group with this property must be solvable. Formulas for the curvature tensor on M are established and used to show that the Lie algebra of any such group S must satisfy a number of structural conditions. Conversely, given a Lie algebra < satisfying these conditions and any member of an easily constructed family of inner products on i, a metric deforma- tion argument is used to obtain a modified inner product which gives rise to a left invariant Riemannian structure with negative curvature on the associated simply connected Lie group. 1. Introduction. This paper was motivated by the following problem: Which connected Lie groups admit a left invariant Riemannian metric with nega- tive (sectional) curvature? We emphasize that throughout the paper, we under- stand "negative" to mean "less than or equal to zero". Since the property in question is not sensitive to groups linked by a local isomorphism, we deal primar- ily with simply connected groups. Results of J. A. Wolf (13) and E. Heintze (4) show that the above problem is closely linked with the classification of connected, homogeneous Riemannian manifolds with negative curvature. Indeed, if M is such a manifold and if M is simply connected, then M is isometric to a solvable Lie group endowed with a left-invariant metric. In this paper, we give a complete solution to our original problem by show- ing that a necessary and sufficient condition for a group to have the property in question is that its Lie algebra be what we call an "NC algebra". Roughly speak- ing, the crucial properties of an NC algebra $ are that in addition to being solv- able, e must contain an abelian subalgebra a complementary to the derived

Kinematics of tensor fields in hyperspace. II

Abstract: Various kinematical relations, holding between hypersurface projections of spacetimetensor fields in an arbitrary Riemannian spacetime, are studied in terms of differential geometry in hyperspace. A criterion is given that a collection of hypertensor fields is generated by the projections of a single spacetimetensor field intersected by the embeddings. From here, it is shown that the super‐Hamiltonian of an arbitrary tensor field splits into two parts, H φ ↑ and H φ −, H φ ↑ being local in the field momenta and H φ − containing their first derivatives. The form of H φ − for an arbitrary tensor field is determined from the field behavior under hypersurface tilts. The kinematical equations for the intrinsic metric and the extrinsic curvature are written in a quasicanocical form, and their connection with the closing relations for the gravitational super‐Hamiltonian is exhibited. The conservation laws of charge, energy and momentum, and the contracted Bianchi identities, are written as hypertensor equations.

The Scalar Curvature

Abstract: We shall deal with some problems concerning the scalar curvature of compact riemannian manifolds. In particular we shall deal with the problem of Yamabe: Does there exist a conformal metric for which the scalar curvature is constant? And also problems posed by Chern, Nirenberg and others. All these problems are almost entirely solved, however there remain some open questions (see the conjectures).

On asymptotically flat space-times

Abstract: The allowed asymptotic behavior of the Ricci tensor is determined for asymptotically flat space-times. With the aid of Penrose's conformai technique the asymptotic behavior of the components of the metric tensor, Weyl tensor, and spin coefficients in a suitable frame is calculated for such a space-time. For Einstein-Maxwell space-times these results reduce to those of Exton, Newman, Penrose, Unti, and Kozarzewski.

Gauge theory of Poincaré symmetry

Abstract: The Einstein-Cartan theory of gravitation is shown to be the unique gauge theory of Poincare symmetry as far as one chooses the Lagrangian to be the lowest possible combination in field strengths. Kibble's derivation of the theory is reformulated and refined in the fiber-bundle picture. The gauge potentials of the Lorentz subgroup are identified as the local affine-connection coefficients which in general allow torsion, and the field strengths of this subgroup are identified as the curvature tensor of the corresponding Riemann-Cartan geometry. The spin current of fermion fields creates the torsion of the geometry. (AIP)

Positive sectional curvatures does not imply positive Gauss-Bonnet integrand

Abstract: An example is given, in dimension six, of a curvature tensor having positive sectional curvatures and negative Gauss-Bonnet integrand. A large class of questions in differential geometry involves the relationship between the topology and the geometry of a compact Riemannian manifold. One of these is the Hopf conjecture: If, in even dimensions, the sectional curvatures of such a manifold are positive, then so is the Euler number. The Hopf conjecture is known to be true in dimensions two and four by the following argument (Milnor, unpublished; [2]). One first writes down the Gauss-Bonnet formula, which, in every even dimension, equates the Euler number of the manifold to a certain integral over the manifold, where the integrand involves only the curvature tensor, and that only algebraically. One then shows (in dimensions two and four) that, at each point, positivity of the sectional curvatures implies positivity of this integrand. Most attempts to prove the full Hopf conjecture have been attempts to generalize this argument [1], [3], [4], [5], [6]. Thus, there arises the following, purely algebraic, question: Over a vector space of any even dimension, does a tensor having the symmetries of a curvature tensor and having positive sectional curvatures necessarily have positive Gauss-Bonnet integrand? We here answer this question in the negative. Fix a real, six-dimensional vector space V. A wedge denotes the wedge product, and a star a Hodge star operator.2 Denote by V2 the vector space of second-rank, antisymmetric tensors over V, by V2 its dual (the space of 2-forms over V), and by V22 the vector space of symmetric linear mappings from V2 to V2. We shall make use of the following fact: For any element A of V2, (1) ((A A A)* A (A A A)*)* 9(A A A A A)*A. For A any element of V2, denote by TA the element of V22 with action Received by the editors September 3, 1974. AMS (MOS) subject classifications (1970). Primary 53B20. i Supported in part by the National Science Foundation under contract GP-34721Xi, and by the Sloan Foundation. 2 Our conventions for the star operation are these: For any form A, A* = A; for B a 2-form and C a 4-form, B(C*) = C(B*) = (B A C)* = (B* A C*)*. Note that we introduce no metric on V. ? American Mathematical Society 1976 267 This content downloaded from 157.55.39.17 on Wed, 31 Aug 2016 04:16:18 UTC All use subject to http://about.jstor.org/terms

Zero-mass plane waves in nonzero gravitational backgrounds

Abstract: The mathematical definition of what is intuitively called a ''plane wave'' on the curved background of a black hole is clarified and discussed from the viewpoints of potentials and fields. Because of the long-range Newtonian part of the gravitational field the asymptotic wave fronts of an incident ''plane wave'' (describing a radiative perturbation for a scattering experiment) are distorted in a manner analogous to the wave fronts of an electron beam in the quantum-mechanical Coulomb scattering problem. In addition, the electromagnetic and gravitational fields can be described with either a potential formalism (i.e., the vector potential and the metric perturbation) or a field formalism (i.e., the electromagnetic field tensor and the Riemann tensor). In this paper we present a distorted ''plane wave'' prescription, necessary for the calculation of the scattering cross sections of electromagnetic and gravitational waves off of a black hole, which agrees with the accepted prescription for a massless scalar field and satisfies the intuitive notions of what constitutes a ''plane wave'' in terms of potentials and fields. (AIP)

Comments on the Jordan-Brans-Dicke scalar-field theory of gravitation

Abstract: The Palatini variational principle is applied to the action integral of the Jordan-Brans-Dicke theory of gravitation. An affinity differing from the Christoffel symbols by an additional third-order tensor is obtained. Field equations with the covariant derivatives and the Ricci tensor defined with respect to this affinity are found. Rewriting these equations using the covariant derivative and the Ricci tensor constructed from the Christoffel symbols yields equations physically equivalent to those of Jordan, Brans and Dicke.

On totally real bisectional curvature

Abstract: A Kaehler manifold of dimension > 3 is a complex space form if and only if it has constant totally real bisectional curvature. Goldberg and Kobayashi [2] introduced the notion of holomorphic bisectional curvature on a Kaehler manifold. It is determined by two holomorphic planes. In this paper, we consider a Kaehler manifold with constant totally real (or called antiholomorphic) bisectional curvature, which is determined by an antiholomorphic plane and its image by the complex structure. Namely it is defined by R(X,JX; Y, JY) for a totally real section {X, Y}. A complex space form is a Kaehler manifold of constant holomorphic sectional curvature. It turns out that a complex space form can be characterized by having constant totally real bisectional curvature. 1. Let M be a real 2n-dimensional Kaehler manifold with complex structure J and Riemann metric g. Let R be the curvature tensor field of M. Then we have R(JX,JY) = R(X, Y) and R(X, Y)JZ = JR(X, Y)Z for any vectors X, Y, Z tangent to M. We denote R(X, Y; ZW) by R(X, Y; Z, W) = g(R(X, Y)Z, W). Then the sectional curvature of M determined by orthonormal vectors X and Y is given by K(X, Y) = R(X, Y; Y, X). It is easy to see then K(JX, JY) = K(X, Y), K(X, JY) = K(JX, Y) and R(X, Y; Z, W) = R(JX, JY; Z, W) = R(X, Y; JZ, JW). By a plane section we mean a 2-dimensional linear subspace of a tangent space. A plane section ff is called holomorphic (respectively antiholomorphic or totally real) if J7, = 7T (respectively J7, is perpendicular to 7r). The sectional curvature for a holomorphic (respectively totally real) plane section is called holomorphic (respectively totally real) sectional curvature. A Kaehler manifold of constant holomorphic sectional curvature is called a complex space form. Let X be a unit vector in a holomorphic plane section, then it is clear Received by the editors December 17, 1974 and, in revised form, March 3, 1975. AMS (MOS) subject classifications (1970). Primary 53B35, 53B20; Secondary 53C55.

K-spaces of constant holomorphic sectional curvature

Abstract: In this note we prove the equivalence of the pointwise constancy and the global constancy of the holomorphic sectional curvature of a K-space. A criterion for the constancy of the holomorphic sectional curvature of a K-space is found. It is proved that every proper K-space of constant holomorphic sectional curvature is a six-dimensional orientable Riemannian manifold of constant positive curvature, which is isometric with the six-dimensional sphere in the case of completeness and connectedness.

Differential geometry on Grassmann algebras

Abstract: H. C. Lee [1] developed the analogue of Riemannian geometry on a real symplectic manifold — the fundamental skew two-form taking the place of the symmetric tensor. The usual Riemannian concepts do not adapt themselves very well, thus ‘curvature’ is represented by a tensor of the third rank and ‘Killing's equations’ now involve this ‘curvature tensor’. The immediate reason for this is that otherwise familiar terms appear with the wrong sign. We have found that these unaesthetic features disappear, and formal elegance is marvellously restored, when the manifold is replaced by a Grassmann algebra. The connection with supersymmetry is explained but applications are not reported here.

Conformally flat manifolds and a pinching problem on the Ricci tensor

Abstract: There is a formal similarity between the theory of hypersurfaces and conformally flat d-dimensional spaces of constant scalar curvature provided d > 3. For, then, the symmetric linear transformation field Q defined by the Ricci tensor satisfies Codazzi's equation (Vx Q)Y = (Vy Q)X. This observation leads to a pinching theorem on the length of the Ricci tensor. 1. Statement of results. Recently, one of the authors [1] obtained THEOREM G. Let M be a d-dimensional compact conformallyflat manifold with definite Ricci curvature. If the scalar curvature r is constant and if the square of the length of the Ricci tensor is not greater than r2/(d 1), d > 3, then M is a space of constant curvature. Note that the square length of the Ricci tensor is greater than or equal to r2ld, so the Ricci tensor has been "pinched". In the present paper the following two theorems are proved, the first of which generalizes Theorem G. THEOREM 1. Let M be a d-dimensional compact conformally flat manifold with constant scalar curvature r. If the length of the Ricci tensor is less than r/ d -1, d > 3, then M is a space of constant curvature. THEOREM 2. In a d-dimensional compact conformally flat manifold M, if the length of the Ricci tensor is constant and less than r/ d -1, then M is a space of constant curvature. 2. Conformally flat manifolds. Let M be a Riemannian manifold of dimension d > 3. We cover M by a system of local coordinate neighborhoods (U, xh), and denote by gji, V>, Rk_h, and R1i the Riemannian metric, the operator of covariant differentiation in terms of the Riemannian connection, the curvature tensor and the Ricci tensor, respectively. We say that M is conformally flat if its Riemannian metric is conformally related to a locally Euclidean metric. In a conformally flat manifold, Received by the editors June 9, 1975 and, in revised form, October 15, 1975. AMS (MOS) subject classifications (1970). Primary 53A30, 53B20, 53C20.

On the stress functions in elastodynamics

Abstract: After a brief review on a theory of stress and stress functions in three-dimensional elastostatics, the author attempts to extend his consideration into the problem of elastodynamics. The stress is represented by the Riemann-Christoffel curvature tensor of a four-dimensional Riemannian space having the stress functions as the components of its metric tensor. From this basic recognition, a representation for stresses by ten stress functions is given. As one of the special cases, the expression for the stress the author used in the analysis of stress fields by moving dislocations is derived. Elementary expressions of the forms which are extensions of Morera's and Maxwell's stress functions are also derived from the general principle.

Electric and magnetic gravitational monopoles. I. The equation of motion of poles

Abstract: The existence of a new class of particles, H poles, is tidal forces that govern the behaviour of nearby H poles in gravitational fields is given by the dual of the Riemann tensor. Consequently, the equivalence principle is not satisfied by these H poles. A physical property of conformal transformation is shown to exist, by means of which the H poles are mapped into particles of geodesic motion. Einstein's equations for the gravitational field in a vacuum are shown to have a larger symmetry group.

Differential Geometry of Totally Real Submanifolds

Abstract: Publisher Summary This chapter describes the differential geometry of totally real sub-manifolds It discusses the properties of totally real submanifolds for cases in which the submanifolds are totally geodesic The chapter highlights preliminaries on the real space form, the Weyl conformal curvature tensor, the complex space form, and the Bochner curvature tensor It presents fundamental formulas for a totally real submanifold M n of a Kaehlerian manifold M 2m The chapter discusses the case in which the ambient space M 2m is a complex space form and the case in which the Bochner curvature tensor of the ambient space M 2m vanishes

Metric Curvature, Folding, and Unique Best Approximation

Abstract: In this paper, the concepts of metric curvature and folding of a $C^1 $-representable manifold in a normed linear space are studied. With certain restrictions on the metric curvature and/or folding, one can obtain a neighborhood of unique best approximation from the manifold, and in some cases, the manifold can be shown to be Chebyshev. Several familiar examples, including some classes of $\gamma $-polynomials, are given.

On the linear connection and curvature in Newtonian mechanics

Abstract: The trajectories of a scleronomic, holonomic particle motion in an otherwise general force field are autoparallel curves in a linear connected, symmetric, ’’almost’’ semimetric space. The Riemann–Christoffel curvature tensor and its concomitants belonging to the dynamical affinity are defined, and the physical meaning is discussed.

Statistical geometry and space-time

Abstract: In this paper I try to construct a mathematical tool by which the full structure of Lorentz geometry to space time can be given, but beyond that the background — to speak pictorially — the subsoil for electromagnetic and matter waves, too. The tool could be useful to describe the connections between various particles, electromagnetism and gravity and to compute observables which were not theoretically related, up to now. Moreover, the tool is simpler than the Riemann tensor: it consists just of a setS of line segments in space time, briefly speaking.

On the structure of complete simply-connected Kähler manifolds with nonpositive curvature

Abstract: We prove that a complete simply-connected Kahler manifold with nonpositive sectional curvature is biholomorphic to the complex Euclidean space if the curvature is suitably small at infinity.