On a problem of Osserman in Lorentzian geometry

Garcı́a-Rı́o, Eduardo; Küpeli, Demir N.; Abal, María Elena Vázquez

doi:10.1016/s0926-2245(96)00037-x

Cited by 46 publications

(53 citation statements)

References 20 publications

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“…timelike) Jordan Osserman if the Jordan normal form of J R on the appropriate pseudo-sphere bundle is independent of P . It is known that any global Riemannian (p = 0) Osserman manifold is locally isometric to a rank 1 symmetric space if m = 8, 16 [3,17,18] and that any local Lorentzian (p = 1) Jordan Osserman manifold has constant sectional curvature [1,5]. In the higher signature setting, there exist spacelike and timelike Jordan Osserman manifolds which are not locally homogeneous [2,7].…”

Section: 4mentioning

confidence: 99%

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The spectral geometry of the Weyl conformal tensor

Blažić

Gilkey

Nikčević³

et al. 2005

PDEs, Submanifolds and Affine Differential Geometry

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Abstract. We study when the Jacobi operator associated to the Weyl conformal curvature tensor has constant eigenvalues on the bundle of unit spacelike or timelike tangent vectors. This leads to questions in the conformal geometry of pseudo-Riemannian manifolds which generalize the Osserman conjecture to this setting. We also study similar questions related to the skew-symmetric curvature operator defined by the Weyl conformal curvature tensor.

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Section: 4mentioning

confidence: 99%

“…If V is Lorentzian and spacelike Jordan Osserman, then results of Blažić, Bokan and Gilkey [1] and of García-Río, Kupeli and Vázquez-Abal [5] shows that V has constant sectional curvature.…”

Section: Theorem 22 Assume Either That (M G) Is An Odd Dimensionalmentioning

confidence: 99%

The spectral geometry of the Weyl conformal tensor

Blažić

Gilkey

Nikčević³

et al. 2005

PDEs, Submanifolds and Affine Differential Geometry

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“…Any spacelike or timelike Osserman manifold is necessarily null Osserman; the converse can fail in general -see for example [García-Río et al 1997] in the Lorentzian setting.…”

Section: Introductionmentioning

confidence: 99%

Four-dimensional Osserman metrics of neutral signature

Río¹,

Gilkey²,

Abal³

et al. 2010

Pacific J. Math.

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In the algebraic context, we show null Osserman, spacelike Osserman, and timelike Osserman are equivalent conditions for a model of signature (2, 2). We also classify the null Jordan Osserman models of signature (2, 2). In the geometric context, we show that a pseudo-Riemannian manifold with this signature is null Jordan Osserman if and only if either it has constant sectional curvature or it is locally a complex space form.

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“…al. [8] shows these are equivalent concepts so one simply speaks of an Osserman manifold. It is known [2,8] that any Lorentzian Osserman manifold has constant sectional curvature; thus the geometry is very rigid in this setting.…”

Section: Introductionmentioning

confidence: 99%

“…[8] shows these are equivalent concepts so one simply speaks of an Osserman manifold. It is known [2,8] that any Lorentzian Osserman manifold has constant sectional curvature; thus the geometry is very rigid in this setting. However if p ≥ 2 and q ≥ 2, there are Osserman pseudo-Riemannian manifolds which are not locally homogeneous; see, for example, [3,7].…”

Section: Introductionmentioning

confidence: 99%