Examples of curvature homogeneous Lorentz metrics

Bueken, Peter; Vanhecke, Lieven

doi:10.1088/0264-9381/14/5/004

Cited by 27 publications

(23 citation statements)

References 7 publications

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“…al. [6]; 1-curvature homogeneous manifolds which are not locally homogeneous were constructed by Bueken and Djorić [4] and by Bueken and Vanhecke [5].…”

Section: Previous Resultsmentioning

confidence: 99%

Complete K-Curvature Homogeneous Pseudo-Riemannian Manifolds 0-Modeled on an Indecomposible Symmetric Space

Gilkey¹,

Nikčević²

2005

Topics in Almost Hermitian Geometry and Related Fields

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“…al. [6]; 1-curvature homogeneous manifolds which are not locally homogeneous were constructed by Bueken and Djorić [4] and by Bueken and Vanhecke [5].…”

Section: Previous Resultsmentioning

confidence: 99%

Complete K-Curvature Homogeneous Pseudo-Riemannian Manifolds 0-Modeled on an Indecomposible Symmetric Space

Gilkey¹,

Nikčević²

2005

Topics in Almost Hermitian Geometry and Related Fields

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“…The equivalence theorem for G-structures due to Cartan and Sternberg [21] provides the generalization of these results to the pseudo-Riemannian case. For more details one can see [8].…”

Section: Theoremmentioning

confidence: 99%

“…Anyhow, constancy of curvature invariants in the pseudo-Riemannian case is a necessary condition but not a sufficient one for local homogeneity of a manifold. In the Lorentzian case, examples are provided in [8] where the authors construct examples of 3-and 4-dimensional Lorentzian manifolds which are curvature homogeneous up to order one, but not locally homogeneous. The conformally flat plane metrics and the conditions to be locally homogeneous have been studied in [15].…”

Section: §0 Introductionmentioning

confidence: 99%

A note on the Osserman conjecture and isotropic covariant derivative of curvature

Bokan¹,

Rakic²,

Blažić³

1999

Proc. Amer. Math. Soc.

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Abstract. Let M be a Riemannian manifold with the Jacobi operator, which has constant eigenvalues, independent on the unit vector X ∈ TpM and the point p ∈ M . Osserman conjectured that these manifolds are flat or rankone locally symmetric spaces (∇R = 0). It is known that for a general pseudo-Riemannian manifold, the Osserman-type conjecture is not true and 4-dimensional Kleinian Jordan-Osserman manifolds are curvature homogeneous. We show that the length of the first covariant derivative of the curvature tensor is isotropic, i.e. ∇R = 0. For known examples of 4-dimensional Osserman manifolds of signature (− − ++) we check also that ∇R = 0. By the presentation of a class of examples we show that curvature homogeneity and ∇R = 0 do not imply local homogeneity; in contrast to the situation in the Riemannian geometry, where it is unknown if the Osserman condition implies local homogeneity. §0. Introduction Let (M, g) be a 4-dimensional Kleinian (neutral) manifold, i.e. a pseudo-Riemannian manifold with a metric g of signature (− − ++). We denote its curvature tensor by R. The Jacobi operator R X : Y → R(Y, X)X is a symmetric endomorphism of T p M and K X is its restriction to X ⊥ in T p M . For Riemannian manifolds, Osserman [16], based on joint results with Sarnak [17], has conjectured that if the eigenvalues of the Jacobi operator K X are independent of the choice of unit vectors X ∈ T p M and of the choice p ∈ M , then either M is locally a rank-one symmetric space or M is flat. We have generalized in [3] the Osserman-type condition in the pseudo-Riemannian setup in terms of the Jordan form of K X , that is equivalent, especially for 4-dimensions, to the conditions in terms of the constancy of the minimal polynomial for K X . Namely, M is spacelike (resp. timelike) Jordan-Osserman at p if the Jordan form of K X is independent of X ∈ T p M , g(X, X) = 1 (resp. g(X, X) = −1). If M is spacelike (resp. timelike) Jordan-Osserman at every p ∈ M , one says M is pointwise spacelike (resp. timelike) Jordan-Osserman. If the Jordan form of K X is independent of p ∈ M , then M is spacelike (resp. timelike) Jordan-Osserman.

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“…One has the following family of examples which are Jacobi-Videv and not Einstein. Manifolds in this family have been studied previously in different contexts, see for example [9,10,11,12,13]; we also refer to [14,15] Definition 1.1. Let k ≥ 1, let ℓ ≥ 1, and m = 2k + ℓ.…”

Section: Introductionmentioning

confidence: 99%