Affine Osserman connections and their Riemann extensions

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

doi:10.1016/s0926-2245(99)00029-7

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Introduction3

Consider the Models1

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2005

2018

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Cited by 44 publications

(57 citation statements)

References 14 publications

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“…If (M, ∇) is locally affine homogeneous, necessarily (M, ∇) is affine k-curvature homogeneous for any k. Examples of 2-curvature homogeneous affine manifolds which are not locally affine homogeneous are known; we refer to the discussion in [9,15,16,17,21] for this and related results.…”

Section: Consider the Modelsmentioning

confidence: 99%

Complete k-Curvature Homogeneous Pseudo-Riemannian Manifolds

Gilkey

Nikčević²

2005

Ann Glob Anal Geom

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Section: Consider the Modelsmentioning

confidence: 99%

Complete k-Curvature Homogeneous Pseudo-Riemannian Manifolds

Gilkey

Nikčević²

2005

Ann Glob Anal Geom

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“…Normalizing the length of the tangent vector to be ±1 takes into account the above scaling of the Jacobi operator. Perhaps surprisingly, spacelike Osserman and timelike Osserman are equivalent conditions [García-Río et al 1999;Gilkey 2001].…”

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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“…One says that an affine manifold (M, ∇) is affine Osserman if Spect{J R ∇ (X)} = {0} for any vector X; i.e Spect{J R ∇ (X)} is nilpotent. Affine Osserman manifolds are well-understood in dimension two, due to the fact that an affine manifold is Osserman if and only if its Ricci tensor is skewsymmetric [4,8]. The situation is however more involved in higher dimensions where the skew-symmetric is a necessary (but not a sufficient) condition for an affine manifold to be Osserman [5,6,7].…”

Section: Introductionmentioning

confidence: 99%