Projective structure in 4-dimensional manifolds with metric signature (+,+,−,−)

Wang, Zhixiang; Hall, Graham

doi:10.1016/j.geomphys.2012.12.004

Cited by 21 publications

(24 citation statements)

References 11 publications

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“…The same statement holds for any pseudo-Riemannian manifold. Since curvature operators belong to holonomy algebra, this explains why the full holonomy and full rank of curvature operators do not appear in results regarding relations of holonomy and geodesical equivalence of metrics (see [5,11]).…”

Section: G-invariant Metrics On Homogenous Space G/hmentioning

confidence: 99%

“…where v * denotes a dual form of v with respect to the metric g. If there exists more than one independent parallel vector fields the family of affinely equivalent metrics can be described in a similar way (for example, see [11,Section 5]).…”

Section: Case Of Left Invariant Metrics On a Lie Groupmentioning

confidence: 99%

“…Geodesically equivalent metrics are actively discussed in the realm of general relativity and have a close relation with integrability and superintegrability (see [8] and references therein). Also, there exists a connection between geodesical equivalence and holonomy groups (see for example [4,5,11]).…”

Section: Introductionmentioning

confidence: 99%

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Geodesically equivalent metrics on homogenous spaces

Bokan¹,

Šukilović²,

Vukmirovic³

2018

Czech.Math.J.

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Two metrics on a manifold are geodesically equivalent if sets of their unparameterized geodesics coincide. In this paper we show that if two left G-invariant metrics of arbitrary signature on homogenous space G/H are geodesically equivalent, they are affinely equivalent, i.e. they have the same Levi-Civita connection. We also prove that existence of non-proportional, geodesically equivalent, G-invariant metrics on homogenous space G/H implies that their holonomy algebra cannot be full. We give an algorithm for finding all left invariant metrics geodesically equivalent to a given left invariant metric on a Lie group. Using that algorithm we prove that no two left invariant metric, of any signature, on sphere S 3 are geodesically equivalent. However, we present examples of Lie groups that admit geodesically equivalent, non-proportional, left-invariant metrics.

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Section: G-invariant Metrics On Homogenous Space G/hmentioning

confidence: 99%

Section: Case Of Left Invariant Metrics On a Lie Groupmentioning

confidence: 99%

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Geodesically equivalent metrics on homogenous spaces

Bokan¹,

Šukilović²,

Vukmirovic³

2018

Czech.Math.J.

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“…There is an alternative approach based on geometric and algebraic considerations for the neutral metric manifolds. This formalism was used in Wang and Hall [11] to classify the holonomy subalgebras in order to study the problem of projectively related manifolds sharing similar holonomy groups.…”

Section: Holonomy Algebras For Neutral Walker Metricsmentioning

confidence: 99%

“…To illustrate this dichotomy in the neutral Ricci-flat VSI Walker metrics, we study two distinct subcases, one which is Walker in general [17,1], and another that is strictly Kundt. We compare these metrics by determining the existence of a null geodesic, expansion-free, shear-free, and vorticity-free vector using the spin-coefficient formalism [9], and then use the Lie algebra classification provided in [7] and [11] to distinguish the two metrics. We show that this classification is well-suited for determining the existence of covariant constant null vectors, recurrent null vectors and more general invariant null distributions, although these comparisons are only helpful for showing when two metrics are not equivalent.…”

Section: Introductionmentioning

confidence: 99%

Mathematical properties of a class of four-dimensional neutral signature metrics

Brooks

Musoke

McNutt

et al. 2015

Journal of Geometry and Physics

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While the Lorenzian and Riemanian metrics for which all polynomial scalar curvature invariants vanish (the VSI property) are well-studied, less is known about the four-dimensional neutral signature metrics with the VSI property. Recently it was shown that the neutral signature metrics belong to two distinct subclasses: the Walker and Kundt metrics. In this paper we have chosen an example from each of the two subcases of the Ricci-flat VSI Walker metrics respectively.To investigate the difference between the metrics we determine the existence of a null, geodesic, expansion-free, shear-free and vorticity-free vector, and classify these spaces using their holonomy algebras. The geometric implications of these algebras are further studied by identifying the recurrent or covariantly constant null vectors, whose existence is required by the holonomy structure in each example. We conclude the paper with a simple example of the equivalence algorithm for these pseudo-Riemannian manifolds, which is the only approach to classification that provides all necessary information to determine equivalence.

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On Holonomy Algebras of Four-Dimensional Generalized Quasi-Einstein Manifolds

Kırık

2018

Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci.

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Projective structure in 4-dimensional manifolds with metric signature (+,+,−,−)

Cited by 21 publications

References 11 publications

Geodesically equivalent metrics on homogenous spaces

Geodesically equivalent metrics on homogenous spaces

Mathematical properties of a class of four-dimensional neutral signature metrics

On Holonomy Algebras of Four-Dimensional Generalized Quasi-Einstein Manifolds

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