Minimal orbits of the isotropy groups of symmetric spaces of compact type

Hirohashi, Daigo; Tasaki, Hiroyuki; Song, Hyunjung; Takagi, Ryoichi

doi:10.1016/s0926-2245(00)00022-x

Cited by 10 publications

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“…In the finite dimensional case minimal orbits of the isotropy representation of symmetric spaces have been systematically studied (e.g. [11], [12]). It may be interesting to ask whether there are similar properties for minimal orbits in the isotropy representations of Kac-Moody symmetric spaces.…”

Section: Open Problemsmentioning

confidence: 99%

Austere and arid properties for PF submanifolds in Hilbert spaces

Morimoto

2020

Differential Geometry and its Applications

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Austere submanifolds and arid submanifolds constitute respectively two different classes of minimal submanifolds in finite dimensional Riemannian manifolds. In this paper we introduce these two notions into a class of proper Fredholm (PF) submanifolds in Hilbert spaces, discuss their relation and show examples of infinite dimensional austere PF submanifolds and arid PF submanifolds in Hilbert spaces. We also mention a classification problem of minimal orbits in hyperpolar PF actions on Hilbert spaces.2010 Mathematics Subject Classification. 53C40.

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Section: Open Problemsmentioning

confidence: 99%

Austere and arid properties for PF submanifolds in Hilbert spaces

Morimoto

2020

Differential Geometry and its Applications

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“…Hirohashi, Tasaki, Song and Takagi proved that, according to the stratification of orbit types, there exists a unique minimal orbit in each orbit type (cf. [HTST,Theorem 3.1]). For commutative Hermann actions which satisfy one of (A), (B) and (C) in Theorem 3.1, Ikawa obtained the same result (cf.…”

Section: Biharmonic Isometric Immersionsmentioning

confidence: 99%

Biharmonic homogeneous submanifolds in compact symmetric spaces and compact Lie groups

Ohno¹,

Sakai²,

Urakawa³

2019

Hiroshima Math. J.

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We give a necessary and sufficient condition for orbits of commutative Hermann actions and actions of the direct product of two symmetric subgroups on compact Lie groups to be biharmonic in terms of symmetric triad with multiplicities. By this criterion, we determine all the proper biharmonic submanifolds in irreducible symmetric spaces of compact type which are singular orbits of commutative Hermann actions of cohomogeneity two. Also, in compact simple Lie groups, we determine all the biharmonic hypersurfaces which are regular orbits of actions of the direct product of two symmetric subgroups which are associated to commutative Hermann actions of cohomogeneity one.

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“…Ã For orbits of s-representations which are minimal submanifolds in the hypersphere, the following theorem is known. THEOREM 3.5 ( [7]). Fix a hypersphere S in m centered at 0.…”

Section: Orbits Of S-representationsmentioning

confidence: 99%

“…Furthermore, typical examples of minimal submanifolds in the hypersphere are given as orbits of s-representations. Hirohashi-Song-Takagi-Tasaki [7] showed that there exists a unique minimal orbit in each strata of the stratification of orbit types. However, in general we can not explicitly point out which orbit among each strata is a minimal submanifold.…”

Section: Introductionmentioning

confidence: 99%

Weakly reflective submanifolds and austere submanifolds

Ikawa¹,

Sakai²,

Tasaki³

2009

J. Math. Soc. Japan

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An austere submanifold is a minimal submanifold where for each normal vector, the set of eigenvalues of its shape operator is invariant under the multiplication by À1. In the present paper, we introduce the notion of weakly reflective submanifold, which is an austere submanifold with a reflection for each normal direction, and study its fundamental properties. Using these, we determine weakly reflective orbits and austere orbits of linear isotropy representations of Riemannian symmetric spaces.

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Minimal orbits of the isotropy groups of symmetric spaces of compact type

Cited by 10 publications

References 0 publications

Austere and arid properties for PF submanifolds in Hilbert spaces

Austere and arid properties for PF submanifolds in Hilbert spaces

Biharmonic homogeneous submanifolds in compact symmetric spaces and compact Lie groups

Weakly reflective submanifolds and austere submanifolds

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