Invariant Einstein Metrics on Some Homogeneous Spaces of Classical Lie Groups

Abstract A Riemannian manifold (M, ρ) is called Einstein if the metric ρ satisfies the condition Ric(ρ) = c · ρ for some constant c. This paper is devoted to the investigation of G-invariant Einstein metrics, with additional symmetries, on some homogeneous spaces G/H of classical groups. As a consequence, we obtain new invariant Einstein metrics on some Stiefel manifolds SO(n)/SO(l). Furthermore, we show that for any positive integer p there exists a Stiefelmanifold SO(n)/SO(l) that admits at least p SO(n)-invariant Einstein metrics.

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Invariant Einstein metrics on $\mathrm{SU}(n)$ and complex Stiefel manifolds

Tohoku Mathematical Journal ◽

10.2748/tmj/1593136818 ◽

2020 ◽

Vol 72 (2) ◽

pp. 161-210 ◽

Cited By ~ 1

Author(s):

Andreas Arvanitoyeorgos ◽

Yusuke Sakane ◽

Marina Statha

Keyword(s):

Einstein Metrics ◽

Stiefel Manifolds

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Invariant K�hler?Einstein metrics on compact homogeneous spaces

Functional Analysis and Its Applications ◽

10.1007/bf01078469 ◽

1986 ◽

Vol 20 (3) ◽

pp. 171-182 ◽

Cited By ~ 37

Author(s):

D. V. Alekseevskii ◽

A. M. Perelomov

Keyword(s):

Einstein Metrics ◽

Homogeneous Spaces

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Bézier Curves on Riemannian Manifolds and Lie Groups With Kinematics Applications

23rd Biennial Mechanisms Conference: Mechanism Synthesis and Analysis ◽

10.1115/detc1994-0173 ◽

1994 ◽

Author(s):

Frank C. Park ◽

Bahram Ravani

Keyword(s):

Riemannian Manifold ◽

Rigid Body ◽

Lie Groups ◽

Riemannian Manifolds ◽

Group Structure ◽

Recursive Algorithm ◽

Bezier Curves ◽

Bézier Curves ◽

Spatial Displacements ◽

Natural Way

Abstract In this article we generalize the concept of Bézier curves to curved spaces, and illustrate this generalization with an application in kinematics. We show how De Casteljau’s algorithm for constructing Bézier curves can be extended in a natural way to Riemannian manifolds. We then consider a special class of Riemannian manifold, the Lie groups. Because of their algebraic group structure Lie groups admit an elegant, efficient recursive algorithm for constructing Bézier curves. Spatial displacements of a rigid body also form a Lie group, and can therefore be interpolated (in the Bezier sense) using this recursive algorithm. We apply this algorithm to the kinematic problem of trajectory generation or motion interpolation for a moving rigid body.

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On Borel–de Siebenthal Representations

International Mathematics Research Notices ◽

10.1093/imrn/rnx304 ◽

2018 ◽

Vol 2019 (15) ◽

pp. 4845-4858

Author(s):

Jing-Song Huang ◽

Yongzhi Luan ◽

Binyong Sun

Keyword(s):

Analytic Continuation ◽

Lie Groups ◽

Discrete Series ◽

Root Systems ◽

Stiefel Manifolds ◽

Simple Lie Groups

AbstractHolomorphic representations are lowest weight representations for simple Lie groups of Hermitian type and have been studied extensively. Inspired by the work of Kobayashi on discrete series for indefinite Stiefel manifolds, Gross–Wallach on quaternonic discrete series and their analytic continuation, and Ørsted–Wolf on Borel–de Siebenthal discrete series, we define and study Borel–de Siebenthal representations (also called quasi-holomorphic representations) associated with Borel–de Siebenthal root systems for simple Lie groups of non-Hermitian type.

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POISSON–LIE GROUPS AND CLASSICAL W-ALGEBRAS

International Journal of Modern Physics A ◽

10.1142/s0217751x92000405 ◽

1992 ◽

Vol 07 (05) ◽

pp. 853-876 ◽

Cited By ~ 33

Author(s):

V. A. FATEEV ◽

S. L. LUKYANOV

Keyword(s):

Field Theory ◽

Conformal Field Theory ◽

Quantum Group ◽

Lie Groups ◽

Group Structure ◽

Two Dimensional ◽

Poisson Structures ◽

Additional Symmetries ◽

Conformal Field ◽

Poisson Lie Groups

This is the first part of a paper studying the quantum group structure of two-dimensional conformal field theory with additional symmetries. We discuss the properties of the Poisson structures possessing classical W-invariance. The Darboux variables for these Poisson structures are constructed.

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