Showing papers on "Ricci decomposition published in 1995"

Einstein structures: Existence versus uniqueness

Abstract: Two well-known questions in differential geometry are “Does every compact manifold of dimension greater than four admit an Einstein metric?” and “Does an Einstein metric of a negative scalar curvature exist on a sphere?” We demonstrate that these questions are related: For everyn≥5 the existence of metrics for which the deviation from being Einstein is arbitrarily small on every compact manifold of dimensionn (or even on every smooth homology sphere of dimensionn) implies the existence of metrics of negative Ricci curvature on the sphereS n for which the deviation from being Einstein is arbitrarily small. Furthermore, assuming either a version of the Palais-Smale condition or the plausible looking existence of an algorithm deciding when a given metric on a compact manifold is close to an Einstein metric, we show for anyn≥5 that: 1) If everyn-dimensional smooth homology sphere admits an Einstein metric thenS n admits infinitely many Einstein structures of volume one and of negative scalar curvature; 2) If every compactn-dimensional manifold admits an Einstein metric then every compactn-dimensional manifold admits infinitely many distinct Einstein structures of volume one and of negative scalar curvature.

Purely magnetic space-times

Abstract: Purely magnetic space–times, in which the Riemann tensor satisfies Rabcdubud=0 for some unit timelike vector ua, are studied. The algebraic consequences for the Weyl and Ricci tensors are examined in detail and consideration given to the uniqueness of ua. Some remarks concerning the nature of the congruence associated with ua are made.

Tensor product surfaces of euclidean plane curves

Abstract: Recently B.Y. CHEN initiated the study of the tensor product immersion of two immersions of a given Riemannian manifold [3]. In [6] the particular case of tensor product of two Euclidean plane curves was studied. The minimal one were classified, and necessary and sufficient conditions for such a tensor product to be totally real or complex or slant were established. In the present paper we study for tensor product of Euclidean plane curves the problem of B.Y. CHEN: to what extent do the properties of the tensor product immersion f ⊗ h of two immersions f, h determines the immersions f, h ? [3]

Finiteness theorem for the ray transform on a Riemannian manifold

Abstract: The ray transform I on a compact Riemannian manifold M with boundary is the operator sending a symmetric tensor field f to the set of integrals of f over all geodesics joining boundary points. A field f is called potential if it can be represented as the symmetric part of the covariant derivative of another tensor field vanishing on the boundary: The main result asserts that the space of potential tensor fields is a subspace of a finite codimension in Ker I if M is simple. A Riemannian manifold is called simple if every two points are joined by a unique geodesic.

Identities of the scalars of the four‐dimensional Riemannian manifold

Abstract: A general algorithm is presented for the calculation of identities relating scalars constructed from appropriate contractions of the Riemann tensor, Ricci tensor, and curvature scalar The algorithm is applied to determining the identities among the quartic scalars

Four-dimensional Einstein-like manifolds and curvature homogeneity

Abstract: The aim of this paper is to classify 4-dimensional Einstein-like manifolds whose Ricci tensor has constant eigenvalues (this being a special kind of curvature homogeneity condition). We give a full classification when the Ricci tensor is of Codazzi type; when the Ricci tensor is cyclic parallel, we classify all such manifolds when not all Ricci curvatures are distinct. In this second case we find a one-parameter family of Riemannian metrics on a Lie groupG as the only possible ones which are irreducible and non-symmetric.

Irrotational dust with a purely magnetic Weyl tensor

Abstract: Considering irrotational pressure free matter, conditions are obtained under which an orthonormal tetrad simultaneously diagonalizes the shear tensor and the electric part of the Weyl tensor. It is shown that such a frame also diagonalises the magnetic part of the Weyl tensor. Using this principal tetrad we show that there are no consistent solutions for irrotational dust with a purely magnetic Weyl tensor.

Hypersurfaces with Parallel Ricci Tensor in Spaces of Constant Curvature

Abstract: Hypersurfaces with parallel Ricci tensor in spaces of constant curvature are classified. The main tool is a generalization of Moore’s decomposition theorem for isometric immersions.

Hypersurfaces of Five Dimensional Space-times

Abstract: It is shown that a space-time hypersurface of a 5-dimensional Ricci-flat space-time has its energy momentum tensor algebrically related to its extrinsic curvature and to the Riemann curvature of the embedding space. It is also seen that the Einstein-Maxwell field does not arise naturally from this geometry, so that a Kaluza-Klein model based on it would require further assumptions.

A Classification of Special Riemannian 3-Manifolds with Distinct Constant Ricci Eigenvalues

Abstract: All Riemannian 3-manifolds with distinct constant Ricci eigenvalues and satisfying some additional geometrical conditions are classified in an explicit form. One obtains locally homogeneous spaces and two different classes of locally non-homogeneous spaces in this way.

Riemannian three‐metrics with degenerate Ricci tensors

Abstract: A complete classification of Riemannian three‐metrics whose Ricci tensor possesses exactly two distinct constant eigenvalues is given: those metrics that admit a G4 as the maximal isometry group belong to either Petrov case IIIq=0, or VII, or VIII; those metrics that admit a G3 as the maximal isometry group belong to either Bianchi class VI0, or VIII, or IX. An explicit coordinate representation is given for all the homogeneous and inhomogeneous solutions.

Lorentzian three-metrics with degenerate Ricci tensors

Abstract: A classification of Lorentzian three‐metrics whose Ricci tensor satisfies Rij=λ1gij+λ2vivj with λ1 and λ2(≠0) constant where vivi=κ(=0 or ±1) is given. An explicit coordinate representation is given for all the metrics that admit a G4 group as their maximal isometry group. Those metrics that admit a G3 as their maximal isometry group belong to either Bianchi class VI0, or VII0, or VIII, or IX when κ ≠ 0, and to either Bianchi class III, or IV, or VI0, VIh, or VIII when κ=0. An explicit coordinate representation is given for all the inhomogeneous solutions in the case κ ≠ 0.

Sasakian manifolds with vanishing C-Bochner curvature tensor

Abstract: As a complex analogue to the Weyl conformal curvature tensor, Bochner and Yano [1], [15] (See also, Tachibana [13]) introduced a Bochner curvature tensor in a Kählerian manifold. Many subjects for vanishing Bochner curvature tensors with constant scalar curvature have been studied by Ki and Kim [6], Kubo [8], Matsumoto [9], Matsumoto and Tanno [11], Yano and Ishihara [16] and so on. One of those, done by Ki and Kim, asserts that the following theorem:

Two-loop vacuum diagrams and tensor decomposition

Abstract: General algorithms for tensor reduction of two-loop massive vacuum diagrams are discussed. Some explicit useful formulae are presented.

Conformal curvature tensor field and spectrum of the laplacian in kaehlerian manifolds

Abstract: Let (M,g) be an m-dimensional compact orientable Riemannian manifold(connected and $C^\infty$) with metric tensor g.

Some remarks for the spectrum of the p-laplacian on sasakian manifolds

Abstract: Let (M,g) be a compact manifold of dimension n with metric tensor g. Let $\Delta^p = d\delta + \deltad$ be the Laplace-Beltrami operator acting on the space of smooth p-forms.

Chemical algebra. VII: ImproperG-weighted metrics of non-compact groups: Lorentz group in the Minkowski space

Abstract: The principles and the generalized equation of chemical algebra is extended to a Minkowskian substrateE endowed with its improper non-definite-positive metric, where the non-compact 6-parameter groupG of the Lorentz transformations operates Given a map μu,u(g) = μ(gu)m(g) onG, a “line element”ds2 is formulated at each point marked by a vectoru Assuming “μ = 1” and “m(g) :≠ 0 ⇒g is a pure Lorentz transformation (without a spatial rotation)”, the isotropic hypothesis (m depends on a single parameter out of three inG) is first studied In general,ds2 does not define a Riemannian manifold unless one additional condition onm is imposed Several relationships are established which are useful for the calculation of the metric tensor and the curvature tensor