Showing papers on "Ricci decomposition published in 1983"
TL;DR: In this paper, it was shown that the original Riemannian metric can be deformed into the constant-curvature metric by requiring that, for t ≥ 0, x ∈ M and g = g(t, x),
Abstract: In [4], R. Hamilton has proved that if a compact manifold M of dimension three admits a C Riemannian metric g0 with positive Ricci curvature, then it also admits a metric g with constant positive sectional curvature, and is thus a quotient of the sphere S. In fact, he shows that the original metric can be deformed into the constant-curvature metric by requiring that, for t ≥ 0, x ∈ M and g = g(t, x),
445 citations
TL;DR: In this paper, the representations of the Riemann and the Weyl tensors through covariant derivatives of third-order potentials are examined in detail, and the possibility of introducing gauges on the potentials is reexamined in connection with the previous result.
Abstract: The representations of the Riemann and the Weyl tensors of a four-dimensional Riemannian manifold through covariant derivatives of third-order potentials are examined in detail. The Weyl tensor always admits a completely general representation whereas the Riemann tensor does not. Nevertheless there exists a class of Riemannian manifolds whose Riemann tensors may be calculated in terms of potentials; in this connection, specific examples are exhibited explicitly. The possibility of introducing gauges on the potentials is reexamined in connection with the previous result. New properties of the representations are also discussed.
84 citations
Journal Article•
TL;DR: In this paper, the conditions générales d'utilisation (http://www.compositio.org/conditions) of the agreement with the Foundation Compositio Mathematica are described.
Abstract: © Foundation Compositio Mathematica, 1983, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
39 citations
TL;DR: In this article, it was shown that for a two-point homogeneous space of constant curvature, the dimension of the space of a Killing tensor field can be computed by products of p Killing 1-forms, by making use of Bott's theorem on holomorphic vector bundles over generalized flag manifolds.
Abstract: A covariant symmetric tensor field ^ on a Riemannian manifold (M, g) is called a Killing tensor field if the symmetrization of the covariant derivative of £ vanishes identically. A Killing tensor field of order 1 is nothing but a Killing 1-form, i.e. a 1-form corresponding to a Killing vector fieldunder the duality by means of the Riemannian metric g. The space K(M, g) of all Killing tensor fieldson (M, g) becomes an algebra by the symmetric product. If the algebra K(M, g) is generated by Killing 1-forms, then the algebra of alllinear differential operators on M which commutes with the Laplacian of (M, g) is generated by Killing vector fields(cf. Theorem 1.1). Sumitomo-Tandai [11] proved the generation of K(Sn, g) by Killing 1-forms for the unit sphere Sn with the standard metric g, by means of the notion of pseudo-connections. This was also proved by C. Tsukamoto by representation theory of compact Lie groups. Sumitomo-Tandai [11] determined moreover the spectrum of the Lichnerowicz Laplacian A (Lichnerowicz [8]) on K(Sn, g), by giving explicitlyprojection operators of K{Sn, g) onto eigenspaces of A. In this paper, for a two-point homogeneous space of constant curvature, we compute the dimension of the space of Killing tensor fieldsspanned by products of p Killing 1-forms, by making use of Bott's theorem (Bott [2]) on holomorphic vector bundles over generalized flag manifolds. Together with the upper bound given by Barbance [1] for the dimension of the space KP(M, g) of Killing tensor fieldsof order p on a general Riemannian manifold (M, g), we prove
29 citations
TL;DR: In this article, a new method for obtaining solutions to Einstein's field equations is presented based on the description of a geometry in terms of the curvature tensor and a number of its covariant derivatives.
Abstract: A new method for obtaining solutions to Einstein's field equations is presented. It is based on the description of a geometry in terms of the curvature tensor and a number of its covariant derivatives. These entities are used instead of the metric. Physical requirements, such as symmetries, are easily imposed in this picture. An example, including the Schwarzschild geometry, is discussed in detail, but no attempts to find new solutions are made.
22 citations
18 citations
TL;DR: In this paper, the complete set of components of the Riemann tensor can be determined by using test particle spin and the corresponding procedure is discussed in detail and compared with three nonlocal methods, which are based on the behavior of classical macroscopic test particles.
Abstract: Quantum mechanically described test particles enable a local measurement of the Riemann tensor via the interaction with the elementary particle spin. The corresponding procedure is discussed in detail. It is compared with three nonlocal methods, which are based on the behavior of classical macroscopic test particles. A central question thereby is if the complete set of components of the Riemann tensor can be determined.
14 citations
TL;DR: In this article, it was shown that a complete hypersurface of non-positive Ricci curvatures in the Euclidean space must be unbounded, under the additional assumption that all sectional curvatures are bounded away from negative infinity.
Abstract: We conjecture that a complete hypersurface of non-positive Ricci curvature in the Euclidean space must be unbounded. We prove this under the additional assumption that all sectional curvatures of the hypersurface are bounded away from negative infinity.
11 citations
TL;DR: In this paper, the combined gravitational-neutrino field equations in general relativity are solved under the following two assumptions: (i) space-time is stationary axially symmetric and the line element of the metric can be put into a canonical form, and (ii) the energy flow vector of the neutrino fields is time-like or null for all observers.
Abstract: The combined gravitational-neutrino field equations in general relativity are solved under the following two assumptions: (i) space-time is stationary axially symmetric and the line element of the metric can be put into a canonical form, (ii) the energy flow vector of the neutrino field is time-like or null for all observers. The resulting metric is uniquely determined and asymptotically non-flat; the Weyl tensor is of Petrov type D, and the Ricci tensor belongs to the class (2T-2S)2 in the Plebanski classification.
6 citations
TL;DR: In this paper, it was shown that the Ricci tensor in the orbit space is always positively defined in Yang-Mills theory, and that the space cannot be considered as compact because it contains infinite-dimensional hypersurfaces.
Abstract: In Yang-Mills theory, it is shown that the Ricci tensor in the orbit space is always positively defined. Nevertheless, the orbit space cannot be considered as compact because it contains infinite-dimensional Euclidean hypersurfaces.
4 citations
TL;DR: In this article, a map carrying irreducible representations of SU(2) into n-fold tensor product spaces of SU (2) is constructed, and it is shown that the multiplicity of an SU( 2) representation in the n- fold tensor space is given by certain Gelfand patterns.
Abstract: A map carrying irreducible representations of SU(2) into n-fold tensor product spaces of SU(2) is constructed. It is shown that the multiplicity of an SU(2) representation in the n-fold tensor product space is given by certain Gelfand patterns.
TL;DR: In this paper, the authors show how flatness and symmetry of an affinely connected manifold relate themselves to the symmetry of the curvature tensor and its covariant derivative in certain slots.
Abstract: Abstract We show how flatness and symmetry of an affinely connected manifold relate themselves to the symmetry of the curvature tensor and its covariant derivative in certain slots. Then we reduce Bianchi first identity to the same form as when the connexion is symmetric; even when the connexion is non-symmetric by considering other conditions.
TL;DR: In this article, the covariant derivative of the Weyl projective curvature tensor vanishes, and a relation between the Brans-Dicke space-time and a space time of constant curvature is discussed.
Abstract: A study of the Brans-Dicke theory in vacuum shows that one of the Brans-Dicke equations is represented in terms of the Riemann curvature tensor, the Weyl projective curvature tensor and its covariant derivative. Studying the case that the covariant derivative of the Weyl projective curvature tensor vanishes, we make a comment on a relation between the Brans-Dicke space-time and a space-time of constant curvature.
TL;DR: In this paper, it was shown that all irreducible representations of a σ-compact Lie group have to be finite dimensional provided that for every π in the reduced dual of G the tensor product π ⊗\(\bar \pi \) has a discrete support.
Abstract: It is shown that all irreducible representations of a σ-compact Lie group G have to be finite dimensional provided that for every π in the reduced dual of G the tensor product π ⊗\(\bar \pi \) has a discrete support.
TL;DR: In this article, a procedure for constructing curvature tensor copies is discussed by using the anholonomic geometrical framework, and the corresponding geometries are compared and the notion of gauge copy elucidated.
Abstract: A procedure for constructing curvature tensor copies is discussed by using the anholonomic geometrical framework. The corresponding geometries are compared and the notion of gauge copy elucidated. An explicit calculation is also made.