Homogeneous geodesics in homogeneous Riemannian manifolds – examples

Kowalski, Oldřich; Nikčević, Stana; Vlášek, Zdeněk

doi:10.1142/9789812792051_0009

Cited by 30 publications

(51 citation statements)

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“…Condition (7) for the operators j (A) can be easily verified from (20). Therefore, relation (19) defines an H-type algebra.…”

Section: Proposition 3 If λ = 0 In (17) the Corresponding Homogeneousmentioning

confidence: 86%

“…where Z is a non-zero vector in the Lie algebra g of G. Homogeneous geodesics have been studied since the sixties by many authors such as Kostant [18], Vingerg [29], Kajzer [15], Szenthe [26], Kowalski [20] and Marinosci [23] among others. The existence problem for homogeneous geodesics was solved recently (and positively) in the most general situation by Dušek, [9].…”

Section: γ (T) = (Exp T Z) · Omentioning

confidence: 99%

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Constancy of Jacobi osculating rank of g.o. spaces of compact and non-compact type

Arias-Marco

Arvanitoyeorgos

Naveira

2011

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“…Condition (7) for the operators j (A) can be easily verified from (20). Therefore, relation (19) defines an H-type algebra.…”

Section: Proposition 3 If λ = 0 In (17) the Corresponding Homogeneousmentioning

confidence: 86%

Section: γ (T) = (Exp T Z) · Omentioning

confidence: 99%

Constancy of Jacobi osculating rank of g.o. spaces of compact and non-compact type

Arias-Marco

Arvanitoyeorgos

Naveira

2011

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“…For results on homogeneous geodesics in homogeneous Riemannian manifolds we refer for example to [3,13,16,18,20]. A homogeneous Riemannian manifold all of whose geodesics are homogeneous is called a Riemannian g.o.…”

Section: Introductionmentioning

confidence: 99%

Homogeneous Geodesics in Homogeneous Affine Manifolds

2009

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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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“…In fact, if Γ is homogeneous with respect to some isometry group K , then it is also homogeneous with respect to any enlarged group of isometries K, but in general the converse does not hold [13]. Homogeneous geodesics of homogeneous Riemannian manifolds have been investigated by many authors.…”

Section: Introductionmentioning

confidence: 99%

“…Homogeneous geodesics of homogeneous Riemannian manifolds have been investigated by many authors. We can refer to [2], [4], [13], [15], [16], [17], [18], for some examples and further references. A fundamental result was obtained by O. Kowalski and J. Szenthe [15], who proved that any homogeneous Riemannian manifold admits at least one homogeneous geodesic.…”

Section: Introductionmentioning

confidence: 99%