Reflections and symmetries in compact symmetric spaces

Chen, Bang‐Yen; Vanhecke, Lieven

doi:10.1017/s000497270002774x

Cited by 5 publications

(6 citation statements)

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“…1. Vanhecke and collaborators [86,87,88,89,90,91] have recently investigated reflections in Riemannian manifolds from a point of view opposite (but complementary) to ours. We start from an involutive isometry and seek to study its fixed-point manifold.…”

Section: A3 Fixed Points Of Isometriesmentioning

confidence: 89%

Wolff-type embedding algorithms for general nonlinear σ-models

et al. 1993

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We study a class of Monte Carlo algorithms for the nonlinear σ-model, based on a Wolff-type embedding of Ising spins into the target manifold M . We argue heuristically that, at least for an asymptotically free model, such an algorithm can have dynamic critical exponent z ≪ 2 only if the embedding is based on an (involutive) isometry of M whose fixed-point manifold has codimension 1. Such an isometry exists only if the manifold is a discrete quotient of a product of spheres. Numerical simulations of the idealized codimension-2 algorithm for the two-dimensional O(4)-symmetric σ-model yield z int,M 2 = 1.5 ± 0.5 (subjective 68% confidence interval), in agreement with our heuristic argument.

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Section: A3 Fixed Points Of Isometriesmentioning

confidence: 89%

Wolff-type embedding algorithms for general nonlinear σ-models

et al. 1993

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“…This should be contrasted with the result proved in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15], characterizing the locally symmetric spaces by the property that the reflections in all geodesics are volume-preserving.…”

Section: Proof Of Theorem 2 and Related Resultsmentioning

confidence: 78%

“…The local generalization of reflections in a line or in a linear subspace has been started in [15] and the main purpose of this paper is to continue this work and to extend it to local reflections in submanifolds of a Riemannian manifold. B. Y. Chen and the second author will treat the role of global reflections and their relation with symmetries in compact symmetric spaces in a forthcoming paper [6].…”

Section: Introductionmentioning

confidence: 99%

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Reflections in submanifolds

Tondeur

Vanhecke

1988

Geom Dedicata

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“…It was known that the polars, meridians and centrosomes play very important roles in the study of compact symmetric spaces as well as of compact Lie groups (cf. [10,11,12,15,17]).…”

Section: 2mentioning

confidence: 99%

The 2-Ranks of Connected Compact Lie Groups

Chen¹

2013

Taiwanese J. Math.

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The 2-rank of a compact Lie group G is the maximal possible rank of the elementary 2-subgroup Z2 × · · · Z2 of G. The study of 2-ranks (and p-rank for any prime p) of compact Lie groups was initiated in 1953 by A. Borel and J.-P. Serre [9]. Since then the 2-ranks of compact Lie groups have been investigated by many mathematician. The 2-ranks of compact Lie groups relate closely with several important areas in mathematics.In this article, we survey important results concerning 2-ranks of compact Lie groups. In particular, we present the complete determination of 2-ranks of compact connected simple Lie groups G via the maximal antipodal sets A2G of G introduced in [16,17].

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Reflections and symmetries in compact symmetric spaces

Abstract: Point symmetries and reflections are two important transformations on a Riemannian manifold. In this article we study the interactions between point symmetries and reflections in a compact symmetric space when the reflections are global isometries.

Cited by 5 publications

References 5 publications

Wolff-type embedding algorithms for general nonlinear σ-models

Wolff-type embedding algorithms for general nonlinear σ-models

Reflections in submanifolds

The 2-Ranks of Connected Compact Lie Groups

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