Geodesic Orbit Metrics in Compact Homogeneous Manifolds With Equivalent Isotropy Submodules

Souris, Nikolaos Panagiotis

doi:10.1007/s00031-017-9464-3

Cited by 21 publications

(15 citation statements)

References 10 publications

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“…Proposition 3.11. ( [1], [32]) A G-invariant metric on G/K, with corresponding metric endomorphism Λ ∈ End(m), is G-geodesic orbit if and only if for any vector X ∈ m there exists a vector…”

Section: 2mentioning

confidence: 99%

Einstein Lie groups, geodesic orbit manifolds and regular Lie subgroups

Souris

2021

Preprint

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We study the relation between two special classes of Riemannian Lie groups G with a left-invariant metric g: The Einstein Lie groups, defined by the condition Ricg = cg, and the geodesic orbit Lie groups, defined by the property that any geodesic is the integral curve of a Killing vector field. The main results imply that extensive classes of compact simple Einstein Lie groups (G, g) are not geodesic orbit manifolds, thus providing large-scale answers to a relevant open question of Y. Nikonorov. Our approach involves studying and characterizing the G × K-invariant geodesic orbit metrics on Lie groups G, where K are certain closed subgroups of G. A key ingredient of our study is the favorable representation-theoretic behaviour of a wide class of subgroups K that we call (weakly) regular. By-products of our work are structural and characterization results that are of independent interest for the classification problem of geodesic orbit manifolds.

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Section: 2mentioning

confidence: 99%

Einstein Lie groups, geodesic orbit manifolds and regular Lie subgroups

Souris

2021

Preprint

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“…Lemma 3.8. ( [19]) Let (G/H, g) be a g.o. space with G compact and with corresponding metric endomorphism A with respect to an Ad-invariant inner product B.…”

Section: Lemma 33 ([17])mentioning

confidence: 99%

“…metrics have been established (e.g. [15], [19]). A general observation is that the existence and the form of the g.o.…”

Section: Introductionmentioning

confidence: 99%

“…12 and X p := X 01 + X 12 = e 12 + X 12 ∈ m 01 + m 12 (since s > 1, m 12 = {0}), where we consider X 12 as arbitrary. Then Equation(19) along with Lemma 4.2 yield0 = λ[a, e 12 ] + λ[a, X 12 ] + 2(λ − µ)f 21 . (20)By the ad(h)-invariance of m ij and the fact that a ∈ h, the first two terms of Equation(20) lie in m 01 and m 12 respectively, while the last term lies in m 01 .…”

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Geodesic orbit metrics in a class of homogeneous bundles over quaternionic Stiefel manifolds

Arvanitoyeorgos,

Souris,

Statha

2021

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Geodesic orbit spaces (or g.o. spaces) are defined as those homogeneous Riemannian spaces (M = G/H, g) whose geodesics are orbits of one-parameter subgroups of G. The corresponding metric g is called a geodesic orbit metric. We study the geodesic orbit spaces of the form (Sp(nSuch spaces include spheres, quaternionic Stiefel manifolds, Grassmann manifolds and quaternionic flag manifolds. The present work is a contribution to the study of g.o. spaces (G/H, g) with H semisimple.

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“…metrics have been established (e.g. [21], [23]). A general observation is that the existence and the form of the g.o.…”

Section: Introductionmentioning

confidence: 99%

Geodesic orbit metrics in a class of homogeneous bundles over real and complex Stiefel manifolds

Arvanitoyeorgos¹,

Souris²,

Statha³

2021

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Geodesic orbit spaces (or g.o. spaces) are defined as those homogeneous Riemannian spaces (M = G/H, g) whose geodesics are orbits of one-parameter subgroups of G. The corresponding metric g is called a geodesic orbit metric. We study the geodesic orbit spaces of the form (G/H, g), such that G is one of the compact classical Lie groups SO(n), U (n), and H is a diagonally embedded product H1 × • • • × Hs, where Hj is of the same type as G. This class includes spheres, Stiefel manifolds, Grassmann manifolds and real flag manifolds. The present work is a contribution to the study of g.o. spaces (G/H, g) with H semisimple.

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Geodesic Orbit Metrics in Compact Homogeneous Manifolds With Equivalent Isotropy Submodules

Cited by 21 publications

References 10 publications

Einstein Lie groups, geodesic orbit manifolds and regular Lie subgroups

Einstein Lie groups, geodesic orbit manifolds and regular Lie subgroups

Geodesic orbit metrics in a class of homogeneous bundles over quaternionic Stiefel manifolds

Geodesic orbit metrics in a class of homogeneous bundles over real and complex Stiefel manifolds

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