Showing papers on "Riemann curvature tensor published in 1970"
TL;DR: In this article, the Bianchi identity is shown to imply that the Misner-Sharp-Hernandez mass function is an integral of two combinations of Einstein's equations for any energymomentum tensor and that mass energy flow is conservative.
Abstract: The mass‐energy of spherically symmetric distributions of material is studied according to general relativity. An arbitrary orthogonal coordinate system is used whenever invariant properties are discussed. The Bianchi identity is shown to imply that the Misner‐Sharp‐Hernandez mass function is an integral of two combinations of Einstein's equations for any energy‐momentum tensor and that mass‐energy flow is conservative. The two mass equations thus found and the mass function provide a technique for casting Einstein's field equations into alternative forms. This mass‐function technique is applied to the general problem of the motion of a perfect fluid and especially to the examination of negative‐mass shells and their relation to singular behavior. The technique is then specialized to the study of a known class of solutions of Einstein's equations for a perfect fluid and to a brief treatment of uniform model universes and the charged point‐mass solution.
207 citations
01 Jan 1970
TL;DR: In this paper, the Curvature tensor has been defined and its properties have been elaborated in terms of physical and geometric properties, including its properties and properties of curvature tensors.
Abstract: In this paper we have defined the Curvature tensor and elaborated its various physical and geometric properties.
110 citations
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TL;DR: In this article, the curvature tensor of Riemannian manifolds is shown to be invariant to curvature in the sense that curvature is an invariant property of the metric.
Abstract: The theorema egregium or, in essence, the fundamental theorem of riemannian geometry asserts that curvature is an invariant of the metric. We ask the converse: how far does curvature determine the metric? Important theorems in this direction are the classical theorems for (embedded) surfaces. More recently there is a local theorem of E. Cartan and its global formulation due to W. Ambrose (cf. [1]). For a different approach see Nomizu and Yano [8]. In these theorems there are non-trivial hypotheses about the curvature tensor. We ask a more naive, but geometrically fascinating question: let (M, g), (M, U) be two Riemann manifolds. Denote the corresponding sectional curvatures by K respectively K. We say, M, M are isocurved if there exists a "sectional-curvature-preserving" diffeomorphism f: M O M, i.e., for every p e M and for every a, a 2-plane section of the tangent space Tp(M), we have K(a) = K(f* a) .
53 citations
TL;DR: It is shown that the Pontrjagin forms of a Riemannian manifold can be expressed uniquely in terms of the conformal curvature tensor: this provides a new proof of Chern and Simons' result.
Abstract: In a recent paper, Chern and Simons proved that the Pontrjagin forms of a Riemannian manifold remain invariant under a conformal deformation. We show that these forms can be expressed uniquely in terms of the conformal curvature tensor: this provides a new proof of their result. Similar techniques can be applied to Euler-Poincare characteristic class, as suggested to me by A. Taub. We obtain the following: If the Weyl tensor of a compact space time is of type III of Bel-Petrov, then it cannot carry a perfect fluid + electromagnetic field.
33 citations
TL;DR: In this article, it was shown that if a Vn admits a parallel vector field, then it will admit groups of curvature collineations (CC) and hence gravitational pp waves admit such groups of symmetries.
Abstract: By definition, a Riemannian space Vn admits a symmetry called a curvature collineation (CC) if the Lie derivative with respect to some vector ξi of the Riemann curvature tensor vanishes. It is shown that if a Vn admits a parallel vector field, then it will admit groups of CC's. It follows that every space‐time with an expansion‐free, shear‐free, rotation‐free, geodesic congruence admits groups of CC's, and hence gravitational pp waves admit such groups of symmetries.
29 citations
26 citations
TL;DR: In this paper, the spin coefficients and the Riemann tensor are represented as linear combinations of the infinitesimal generators of the group SL(2, C) in a way similar to the way Yang and Mills write their dynamical variables in terms of the Pauli spin matrices.
Abstract: We represent the spin coefficients and the Riemann tensor in the form of linear combinations of the infinitesimal generators of the group SL(2, C). This representation is similar to the way Yang and Mills write their dynamical variables in terms of the Pauli spin matrices. The spin coefficients take the role of the Yang‐Mills‐like potentials, whereas the Riemann tensor takes the role of the fields.
20 citations
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TL;DR: A computing system that enables problems of manipulative algebra involving a number of elementary functions to be simply and efficiently programmed for the explicit calculation of the Riemann tensor and associated quantities is described.
Abstract: This paper describes a computing system that enables problems of manipulative algebra involving a number of elementary functions to be simply and efficiently programmed. The system has been designed with particular reference to the problems involved in the explicit calculation of the Riemann tensor and associated quantities.
14 citations
14 citations
01 Jul 1970
14 citations
TL;DR: In this article, a local rigidity theorem for minimal hypersurfaces on a Riemannian manifold of constant curvature has been proposed, where the curvature tensor of the Ricci tensor is defined.
Abstract: [ 1 ] S. S. CHERN, Minimal submanifolds in a Riemannian manifold, Lecture note, 1968.[2] S. S.CHERN, M.P. Do Carmo, S. KOBAYASHI, Minimal submanifolds of a sphere withsecond fundamental form of constant length, to appear.[ 3 ] E. T. DA VIES, On the second and third fundamental forms of a subspace, J. London Math.Soc, 12(1937), 290-295.[ 4 ] H. B. LAWSON, JR, Local rigidity theorems for minimal hypersurfaces, Ann. of Math., 89(1969), 187-197.[ 5 ] K. NOMIZU, On hypersurfaces satisfying a certain condition on the curvature tensor,Tδhoku Math. J., 20(1968), 46-59.[ 6 ] T. OTSTJKI, Minimal hypersurfaces in a Riemannian manifold of constant curvature, toappear.[7] J.SlMONS, Minimal varieties in riemannian manifolds, Ann. of Math., 88(1968), 62-105.[8] S. TANNO, Hypersurfaces satisfying a certain condition on the Ricci tensor, Tόhoku Math.J., 21(1969), 297-303.[9] S. TANNO, T. TAKAHASHI, Some hypersurfaces of a sphere, Tόhoku Math. J., 22(1970),212-219.[10] Y. TOMONAGA, Pseudo-Jacobi fields on minimal varieties, Tohoku Math. J., 21(1969),539-547.MATHEMATICAL INSTITUTETOHOKU UNIVERSITYS::NDAI, JAPAN
TL;DR: In this article, the authors generalized the Lipschitz-Killing curvature to a complete, simply-connected Riemannian manifold with non-positive sectional curvature, and then defined the total absolute curvature as
Abstract: Willmore and Saleemi [12] had generalized Chern-Lashof s results by definingthe total curvature of an orientable manifold immersed in a Riemannian manifold,but unfortunately, the results contained mistakes, and hence they are false.The object of this paper is to generalize the Lipschitz-Killing curvature to themanifolds immersed in a complete, simply-connected Riemannian manifold withnon-positive sectional curvature, and then to define the total absolute curvature as
TL;DR: In this article, the behavior of asymptotically flat gravitational fields in the framework of general relativity is studied by the use of tetrad formalism, and the form of the peeling theorem in the above-mentioned coordinates for an arbitrary null tetrad is derived.
Abstract: The behaviour of asymptotically flat gravitational fields in the framework of general relativity is studied by the use of tetrad formalism. For this, a system of coordinates u, r, H and 0 is used, such that at spatial infinity u = const. is a null hypersurface and r, 0 and 5 reduce to the usual spherical polar coordinates. A set of four vectors (a tetrad) is also chosen with the only restriction that they are everywhere null. The metric tensor and the four vectors are expanded in inverse powers of r; the rotation coefficients and the tetrad components of the Riemann tensor are then calculated in a similar expansion; and the first two terms in the expansion beyond their values for a flat space are retained. The field equations in these approximations are derived explicitly and their effect on the expansion of the tetrad components of the Riemann tensor is studied; and the total energy and linear momentum are examined. In this paper three main results are derived: (i) the form of the peeling theorem in the above-mentioned coordinates for an arbitrary null tetrad; (ii) the generalized expression for the news function of the field; (iii) a simple criterion for recognizing certain classes of non-radiating fields. 1. INTRODIUCTION During the last decade a great deal of work has been done on asymptotically flat spaces in general relativity. Bondi, van der Burg & Metzner (i962) investigated the case of a gravitational field with an axis and a plane of symmetry while Sachs (i962) investigated the general case of a field with no symmetries. Among their results, two important ones were the mass-loss formula and the so-called peeling theorem. Using a slightly different approach, Newman & Penrose (I962) and Newman & Unti (I962) derived the peeling theorem from different assumptions and analysed in more detail the field at infinity. And finally, Newman & Penrose (I965) discovered ten absolute constants of the field with physical meanings still unknown. All the foregoing investigations were carried out in a system of coordinates
TL;DR: In this article, a temporal metric tensor is defined by combining the sound-speed function with the spatial metric tensors for a Riemannian space, which leads to the consideration of geodesic deviation and its relation to three-dimensional spreading loss.
Abstract: A temporal metric tensor is defined by combining the sound‐speed function with the spatial metric tensor for a Riemannian space. Fermat's principle implies that spatial rays are temporal geodesics. Ray equations generalized to Riemannian spaces are shown to be temporal geodesic equations expressed in spatial terms. This geometric derivation leads to the consideration of geodesic deviation and its relation to three‐dimensional spreading loss. Previous results [E. S. Eby, “Frenet Formulation of Three‐Dimensional Ray Tracing,” J. Acoust. Soc. Amer. 42, 1287–1297 (1967)] are generalized to Riemannian spaces, and tensor expressions are derived for ray curvature and torsion.
TL;DR: In this article, the authors considered surfaces of negative extrinsic curvature in a Riemannian space with nonpositive curvature and proved that the following inequality holds on a surface which is complete in the sense of the intrinsic metric: \frac{1}{\sqrt{3}}; $ SRC=http://ej.iop.org/images/0025-5734/12/2/A10/tex_sm_923_img1.
Abstract: We consider surfaces of negative extrinsic curvature in a Riemannian space with nonpositive curvature. We prove that the following inequality holds on a surface which is complete in the sense of the intrinsic metric: \frac{1}{\sqrt{3}}; $ SRC=http://ej.iop.org/images/0025-5734/12/2/A10/tex_sm_923_img1.gif/> here F is the surface being considered, (Ke is the extrinsic curvature of F) and Λ and λ are the maximum and minimum of the Riemannian curvature of the space R at a given point. This theorem generalizes a theorem of Efimov concerning q-metrics. We give an example of a surface for which q=4.5. Bibliography: 8 items.
TL;DR: In this article, Loos et al. showed that if a certain algebraic restriction involving the Riemann tensor, say R, is satisfied, then there is a one-to-one relation between the spin connection sΓμab and the parameters of a riemannian event connection Γμκλ in a neighborhood of an assumed solution of the equations which establish the correspondence.
Abstract: If parameters of a spin connection have a holonomy group which is the Lorentz group or a subgroup thereof and if a certain algebraic restriction involving the Riemann tensor, say R, is satisfied, then there is a one‐to‐one relation between the spin connection sΓμab and parameters of a Riemannian event connection Γμκλ, in a neighborhood of an assumed solution of the equations which establish the correspondence [see H G Loos, Ann Phys 25, 91 (1963)] We have, by the use of anholonomic bases in the Riemann space, simplified the equations which determine the map and have shown that if the algebraic condition R is satisfied, a one‐to‐one correspondence holds also in the large The same method, involving anholonomic bases, has also been used to study the map between gauge potentials acting in the internal space formed by generators of local Lorentz transformations and parameters of connection of a Riemannian event space An analogous theorem is proved which asserts that if R is satisfied, then the correspon
TL;DR: On the hypersurface z = 0 the Riemannian tensor Riklm and the Ricci tensor Rkl become delta-like infinite as discussed by the authors, and the first derivations of the metric tensor gik have essential discontinuities.
Abstract: The Einstein-Rosen matter is given by a space-time V4 in which the domains z 0 are isometric But, on the hypersurface z = 0 the Riemannian tensor Riklm and the Ricci tensor Rkl become delta-like infinite On this surface the first derivations of the metric tensor gik have essential discontinuities Therefore, it is impossible to cover the small domain –e ≤ z ≤ +e with one coordinate system of the class C2
01 Jan 1970
TL;DR: SINHA and Singh as discussed by the authors defined the recurrent curvature tensor fields of second order and studied the properties of recurrence tensor field and Finsler spaces in the Ricci recurrent spaces.
Abstract: By B. B. SINHA and S. P. SINGH (Received Feb. 20, 1970) Summary. Chaki and Roychowdhary have studied the Ricci recurrent spaces of second order in the Riemannian Geometry $[$ 1 $]$ . The object of present paper is to define the recurrent curvature tensor fields of second order and to study the properties of recurrence tensor field and the curvature tensor fields in the Finsler spaces.
TL;DR: Theorems on the structure of the fundamental group of a compact riemannian manifold of non-positive curvature are proved, including a conjecture of J. Wolf.
Abstract: We prove theorems on the structure of the fundamental group of a compact riemannian manifold of non-positive curvature. In particular, a conjecture of J. Wolf [J. Differential Geometry, 2, 421-446 (1968)] is proved.
TL;DR: In this article, the same condition turns the conservation law (which is a consequence of the field equations) into an identity and that the Rainich decomposition of Riemann's curvature tensor gives only one component.
Abstract: The equations of the de Broglie wave field (field equations) [J Kulhanek, Nuovo Cimento Supp 4, 172 (1966)] under special conditions require a very particular geometry together with a specific interpretation of the curvature scalar The purpose of the present paper is to show that the same condition turns the conservation law (which is a consequence of the field equations) into an identity and that the Rainich [Nature 115, 498 (1925)] decomposition of Riemann's curvature tensor gives only one component