Showing papers on "Riemann curvature tensor published in 1969"

Curvature Collineations: A Fundamental Symmetry Property of the Space‐Times of General Relativity Defined by the Vanishing Lie Derivative of the Riemann Curvature Tensor

Abstract: A Riemannian space Vn is said to admit a particular symmetry which we call a ``curvature collineation'' (CC) if there exists a vector ξi for which £ξRjkmi=0, where Rjkmi is the Riemann curvature tensor and £ξ denotes the Lie derivative. The investigation of this symmetry property of space‐time is strongly motivated by the all‐important role of the Riemannian curvature tensor in the theory of general relativity. For space‐times with zero Ricci tensor, it follows that the more familiar symmetries such as projective and conformal collineations (including affine collineations, motions, conformal and homothetic motions) are subcases of CC. In a V4 with vanishing scalar curvature R, a covariant conservation law generator is obtained as a consequence of the existence of a CC. This generator is shown to be directly related to a generator obtained by means of a direct construction by Sachs for null electromagnetic radiation fields. For pure null‐gravitational space‐times (implying vanishing Ricci tensor) which admit CC, a similar covariant conservation law generator is shown to exist. In addition it is found that such space‐times admit the more general generator (recently mentioned by Komar for the case of Killing vectors) of the form (−g Tijkmξiξjξk);m=0, involving the Bel‐Robinson tensor Tijkm. Also it is found that the identity of Komar, [−g(ξi;j−ξj;i)];i;j=0, which serves as a covariant generator of field conservation laws in the theory of general relativity appears in a natural manner as an essentially trivial necessary condition for the existence of a CC in space‐time. In addition it is shown that for a particular class of CC,£ξK is proportional to K, where K is the Riemannian curvature defined at a point in terms of two vectors, one of which is the CC vector. It is also shown that a space‐time which admits certain types of CC also admits cubic first integrals for mass particles with geodesic trajectories. Finally, a class of null electromagnetic space‐times is analyzed in detail to obtain the explicit CC vectors which they admit.

On conformal killing tensor in a riemannian space

Abstract: where pc is a certain vector field. Because we can easily show that a conformal Killing tensor in this sense is a Killing tensor, i.e., we have pc = 0. Thus this definition of conformal Killing tensor is meaningless. In this paper we shall define a conformal Killing tensor in another way and generalize some results about a conformal Killing vector to the conformal Killing tensor. The definition which we shall adopt is suggested by the following fact. A parallel vector field in the Euclidean space E induces a

Curvature and the Petrov Canonical Forms

Abstract: The Petrov classification for the curvature tensor of an Einstein space M4 is related to the critical‐point theory of the sectional‐curvature function σ, regarded as a function on the manifold of nondegenerate tangent 2‐planes at each point of the space. It is shown that the Petrov type is determined by the number of critical points. Furthermore, all the invariants in the canonical form can be computed from a knowledge of the critical value and the Hessian quadratic form of σ at any single critical point.

The exact static exterior and interior metric of a thick plane plate

Abstract: The exact static exterior and interior solution of Einstein's equations for a plane thick disk is obtained. It has been shown that the corresponding energy-momentum tensor fulfils the generalized O'Brien-Synge junction conditions. The correspondence of this tensor to the surface energy-momentum tensor of a thin plane plate is demonstrated. The disk being neither very thick nor very dense, a connection with the Newtonian theory has been obtained.

An Observable Peculiarity of the Brans-Dicke Radiation Zone

Abstract: It is pointed out that the Brans-Dicke theory predicts a transverse spin-0 component of the $O({r}^{\ensuremath{-}1})$ radiative Riemann tensor. Unlike the conventional spin-2 radiative Riemann tensor, monopole oscillations may serve as a source. Under favorable conditions, neutron star formation within our galaxy will be detectable by Weber's gravitational-wave receiver.

Sources of curvature of a vector field

Abstract: It is known that for a vector field in three-dimensional space we can introduce the concepts of curvature and mean curvature. In the present article we derive integral formulas for these concepts; these formulas allow us to decide whether a vector field has, for example, singularities in a domain. We explain the influence of the modulus of the curvature of a vector field on the magnitude of its nonholonomity. We also consider the question of the influence of the curvature of a family of surfaces on the distortion of the enveloping space for a given size of domain. Bibliography 5 items.

Complex spaces with locally product metrics: general theory

Abstract: Locally product complex spaces are introduced with particular reference to the Calculus of Variations on complex manifolds. The goedesics of such spaces possess unusual features associated with the fact that many of the connection coefficients are tensorial in character. A restricted partial covariant derivative is introduced together with a curvature tensor. The latter is found to be invariant under a large class of gauge-like transformations. This invariance property leads naturally to the introduction of almost totally decomposable complex spaces. Necessary and sufficient conditions for a locally product complex Riemannian space to be almost totally decomposable are discussed. The counterpart of the Einstein tensor is also exhibited.

Infinitely small transformations preserving the curvature of a Riemannian space

Abstract: The necessary and sufficient conditions for tensor character are obtained, for which an infinitely small transformation of the space Vn preserves its Riemannian curvature for any two-dimensional area. It is proved that for n>3 the subprojective spaces of the exceptional case, satisfying a certain condition, and only they, permit nontrivial, infinitely small conformal transformations preserving the Riemannian curvature of each two-dimensional area.

A Contribution to the Vector and Tensor Analysis I

Abstract: -In a previous paper [2] we developed the basic features of the vector and tensor analysis under the assumption that there is an n -dimensional vector space at every point of an m -dimensional parameter manifold. Here we make the final step to connect the vector spaces X(P), P ∈ Ω with the parameter manifold as far as possible. Such a manifold with a structure determined by the fundamental tensor and by the connection coefficients may be called a space. We therefore require the dimension m of the parameter manifold and the dimension n of the vector spaces X(P), P ∈ Ω to be equal, m = n Naturally, the results obtained for the case m not necessarily equal to n [2], when adapted to the case m = n remain conserved, while some new results and relations, characteristic of the case m = n, will be derived.

Generalisations of Gauss-Codazzi Equations for Second of Cartan's Curvature Tensors in a Hypersurface of a Finsler Space

Abstract: In a recent paper Rund [2] obtained the Gauss- Codazzi equations for a hypersurface of a Finsler space in respect of the tensor usually denoted by K αβγδ , this tensor representing the generalization of the curvature tensor of Riemannian geometry.