Nilpotent Szabó, Osserman and Ivanov-Petrova pseudo-Riemannian manifolds

Fiedler, Bernd; Gilkey, Peter Β.

doi:10.1090/conm/337/06051

Cited by 10 publications

(15 citation statements)

References 13 publications

(16 reference statements)

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“…Since S(v − ) is self-adjoint with respect to a definite metric on E λ , S(v − ) is diagonalizable on E λ . This shows S(v − ) is Jordan simple and establishes assertion (2) and completes the proof of Theorem 1.5.…”

Section: Timelike and Spacelike Jordan Szabó Tensorssupporting

confidence: 73%

“…Proof of Theorem 1.6 (2). Let V be a vector space of signature (p, q) where q ≡ 2 mod 4 and where p < q − 1.…”

Section: Jordan Szabó Tensorsmentioning

confidence: 98%

“…Let V C := V ⊗ C be the complexification of V . We extend the inner product and the tensor R to be complex multilinear on 2 have the same spectrum. This completes the proof of assertion (1) of Theorem 1.3 by establishing the identity:…”

Section: Spacelike Szabó Tensorsmentioning

confidence: 99%

“…If p ≥ 2 and if q ≥ 2 , then there non-trivial nilpotent Szabó pseudo-Riemannian manifolds [2,6]. However, there are as yet no known examples of spacelike Jordan Szabó tensors R, where R = 0, and we conjecture none exist.…”

Section: Introductionmentioning

confidence: 97%

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Jordan Szabó algebraic covariant derivative curvature tensors

Gilkey¹,

Ivanova²,

Stavrov³

2003

Recent Advances in Riemannian and Lorentzian Geometries

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We show that if R is a Jordan Szabó algebraic covariant derivative curvature tensor on a vector space of signature (p, q), where q ≡ 1 mod 2 and p < q or q ≡ 2 mod 4 and p < q − 1, then R = 0. This algebraic result yields an elementary proof of the geometrical fact that any pointwise totally isotropic pseudo-Riemannian manifold with such a signature (p, q) is locally symmetric.Since A λ = Aλ, E λ = Eλ. Both A and A λ preserve each generalized eigenspace E λ . The operator A is said to be Jordan simple if A λ = 0 on E λ for all λ. We say λ is an eigenvalue of A if dim{E λ } > 0; the spectrum Spec {A} ⊂ C is the set of complex eigenvalues of A.Lemma 1.1. Let A be a self-adjoint linear map of a vector space of signature (p, q).(1) We may decompose V = ⊕ Im (λ)≥0 E λ .(2) We have E λ ⊥ E µ if λ = µ and λ =μ.(3) The spaces E λ inherit non-degenerate metrics of signature (p λ , q λ ). (4) If λ / ∈ R, then p λ = q λ .

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Section: Timelike and Spacelike Jordan Szabó Tensorssupporting

confidence: 73%

“…Proof of Theorem 1.6 (2). Let V be a vector space of signature (p, q) where q ≡ 2 mod 4 and where p < q − 1.…”

Section: Jordan Szabó Tensorsmentioning

confidence: 98%

Section: Spacelike Szabó Tensorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

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Jordan Szabó algebraic covariant derivative curvature tensors

Gilkey¹,

Ivanova²,

Stavrov³

2003

Recent Advances in Riemannian and Lorentzian Geometries

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“…Our aim in this work is twofold. First, we investigate IP metrics in signature (2,2), where only partial results are known [3,7,13], with special attention to Walker metrics. Our second purpose is to study locally symmetric and locally homogeneous affine surfaces with symmetric and degenerate Ricci tensor.…”

Section: Introductionmentioning

confidence: 99%

Riemann Extensions of Torsion-Free Connections with Degenerate Ricci Tensor

2010

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The global geometry of Riemannian manifolds with commuting curvature operators

Brozos‐Vázquez

Gilkey

2006

J.fixed point theory appl.

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Abstract. We give manifolds whose Riemann curvature operators commute, i.e. which satisfy R(x 1 , x 2 )R(x 3 , x 4 ) = R(x 3 , x 4 )R(x 1 , x 2 ) for all tangent vectors x i in both the Riemannian and the higher signature settings. These manifolds have global geometric phenomena which are quite different for higher signature manifolds than they are for Riemannian manifolds. Our focus is on global properties; questions of geodesic completeness and the behaviour of the exponential map are investigated.

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Nilpotent Szabó, Osserman and Ivanov-Petrova pseudo-Riemannian manifolds

Abstract: We exhibit pseudo Riemannian manifolds which are Szabó nilpotent of arbitrary order, or which are Osserman nilpotent of arbitrary order, or which are Ivanov-Petrova nilpotent of order 3. Date: Version W06v6c.tex last changed 05 November 2002.

Cited by 10 publications

References 13 publications

Jordan Szabó algebraic covariant derivative curvature tensors

Jordan Szabó algebraic covariant derivative curvature tensors

Riemann Extensions of Torsion-Free Connections with Degenerate Ricci Tensor

The global geometry of Riemannian manifolds with commuting curvature operators

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