Almost g.o. spaces in dimensions 6 and 7

Dušek, Zdeněk

doi:10.1515/advgeom.2009.006

Cited by 7 publications

(11 citation statements)

References 13 publications

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“…Recall the conjecture in the Introduction. The geodesic lemma allows us to rephrase its statement: Conjecture 1 [8]. If an homogeneous space G/H is an almost g.o.…”

Section: An Almost Go Space Whose Null Homogeneous Geodesics Requirmentioning

confidence: 99%

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Homogeneous geodesics in pseudo-Riemannian nilmanifolds

Barco

2016

Advances in Geometry

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Abstract. We study the geodesic orbit property for nilpotent Lie groups N when endowed with a pseudo-Riemannian left-invariant metric. We consider this property with respect to different groups acting by isometries. When N acts on itself by left-translations we show that it is a geodesic orbit space if and only if the metric is bi-invariant. Assuming N is 2-step nilpotent and with non-degenerate center we give algebraic conditions on the Lie algebra n of N in order to verify that every geodesic is the orbit of a one-parameter subgroup of N ⋊ Auto(N ). In addition we present an example of an almost g.o. space such that for null homogeneous geodesics, the natural parameter of the orbit is not always the affine parameter of the geodesic.

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“…Recall the conjecture in the Introduction. The geodesic lemma allows us to rephrase its statement: Conjecture 1 [8]. If an homogeneous space G/H is an almost g.o.…”

Section: An Almost Go Space Whose Null Homogeneous Geodesics Requirmentioning

confidence: 99%

“…To conclude this work we consider the following conjecture posed by Dušek in [8]: Conjecture 1: If an homogeneous pseudo-Riemannian space is a geodesic orbit space or an almost geodesic orbit space, then the parameter of the oneparameter group whose orbit is a null geodesic is an affine parameter for the geodesic.…”

Section: Introductionmentioning

confidence: 99%

Homogeneous geodesics in pseudo-Riemannian nilmanifolds

Barco

2016

Advances in Geometry

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“…spaces and even for all known almost g.o. spaces [5,8], we have observed that each homogeneous geodesic is expressed through the affine parameter.…”

Section: Conventionmentioning

confidence: 99%

“…In these papers, each geodesic vector X is characterized by the formula (1.2). See also [4,5,7,8]. A rigorous mathematical proof of this characterization is given in [7]:…”

Section: Introductionmentioning

confidence: 99%

Homogeneous Geodesics in Homogeneous Affine Manifolds

2009

Self Cite

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For studying homogeneous geodesics in Riemannian and pseudoRiemannian geometry (on reductive homogeneous spaces) there is a simple algebraic formula which involves the reductive decomposition g = h +m of the Lie algebra g of the isometry group G and the scalar product on m induced by the metric. In the affine differential geometry, there is not such a universal formula. In the present paper, we propose a simple method of investigation of affine homogeneous geodesics. As an application, we describe homogeneous geodesics for homogeneous affine connections in dimension 2 and we find families of affine g.o. spaces in dimension 2. We also solved the problem of the canonical re-parametrization of affine homogeneous geodesics. Mathematics Subject Classification (2000). 53B05, 53C30.

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“…Furthermore, it is conjectured that a pseudo-Riemannian g.o. space with non-compact isotropy group should be naturally reductive [9].…”

Section: Introductionmentioning

confidence: 99%