Geodesic orbit spheres and constant curvature in Finsler geometry

Xu, Ming

doi:10.1016/j.difgeo.2018.07.002

Cited by 12 publications

(11 citation statements)

References 31 publications

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“…The case (1.5), (2.1), (2.2) and (2.3) are ϕ-symmetric spaces [11][15], i.e., weakly symmetric with respect to the action of G × SO (2). By Lemma 4.2 in [17] and Corollary 6.1, standard homogeneous (α 1 , α 2 )-metrics on these coset spaces are G × SO(2)invariant. So they satisfy both the G × SO(2)-weakly symmetric property and the G-g.o.…”

Section: Besides That We Havementioning

confidence: 99%

Standard homogeneous $(α_1,α_2)$-metrics and geodesic orbit property

Zhang¹,

Xu²

2019

Preprint

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In this paper, we introduce the notion of standard homogeneous (α 1 , α 2 )metrics, as a natural non-Riemannian deformation for the normal homogeneous Riemannian metrics. We prove that with respect to the given bi-invariant inner product and orthogonal decompositions for g, if there exists one generic standard g.o. (α 1 , α 2 )-metric, then all other standard homogeneous (α 1 , α 2 )-metrics are also g.o.. For standard homogeneous (α 1 , α 2 )-metrics associated with a triple of compact connected Lie groups, we can refine our theorem and get some simple algebraic equations as the criterion for the g.o. property. As the application of this criterion, we discuss standard g.o. (α 1 , α 2 )-metric from H. Tamaru's classication work, and find some new examples of non-Riemannian g.o. Finsler spaces which are not weakly symmetric. On the other hand, we also prove that all standard g.o. (α 1 , α 2 )metrics on the three Wallach spaces, W 6 = SU (3)/T 2 , W 12 = Sp(3)/Sp(1) 3 and W 24 = F 4 /Spin(8), must be the normal homogeneous Riemannian metrics.

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Section: Besides That We Havementioning

confidence: 99%

Standard homogeneous $(α_1,α_2)$-metrics and geodesic orbit property

Zhang¹,

Xu²

2019

Preprint

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“…In [22], we have classified the geodesic orbit Finsler spheres by the following theorem, which generalizes a theorem of Yu.G. Nikonorov in the Riemannian context [19].…”

Section: Preliminariesmentioning

confidence: 99%

“…Using Theorem 1.1, we can provide a more self contained proof of the following theorem in [22], without using [17] (i.e. Theorem 6.2 in [22]).…”

Section: Corollary 13 Any Reversible Homogeneous Finsler Sphere Withmentioning

confidence: 99%

“…By the classification of geodesic orbit Finsler spheres in [22], Theorem 1.5 can be equivalently stated as the following.…”

Section: Corollary 13 Any Reversible Homogeneous Finsler Sphere Withmentioning

confidence: 99%

“…In Section 2, we summarize some necessary knowledge on Finsler geometry and homogeneous geometry. In Section 3, we prove Theorem 1.1, and sketch an alternative approach proving Theorem 1.5 (Theorem 6.2 in [22]). In Section 3, we discuss the behavior of geodesics on a homogeneous Finsler sphere with K ≡ 1.…”

Section: Introductionmentioning

confidence: 99%

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Reversible homogeneous Finsler metrics with positive flag curvature

Ziller

2016

Forum Mathematicum

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Many of the known results about Riemannian manifolds with positive sectional curvature carry over to the setting of Finsler manifolds with positive flag curvature, essentially using the same proof, see [BCS]. Exceptions are those results that require explicit formulas for the curvature. Part of the difficulty is that sectional curvature needs to be replaced by the flag curvature, which depends not only on the plane, but also on a given vector in the plane. a) Sp(2)/ diag(z, z 3 ) with z ∈ C; (b) Sp(2)/ diag(z, z) with z ∈ C; (c) Sp(3)/ diag(z, z, r) with z ∈ C, q ∈ H; (d) SU(4)/ diag(zA, z,z 3 ) with A ∈ SU(2) and z ∈ C; (e) G 2 / SU(2) with SU(2) the normal subgroup of SO(4) corresponding to the long root.

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