Groups of automorphisms of Lie groups: density properties, bounded orbits, and homogeneous spaces of finite volume

Abstract We provide a general strategy to construct multilinear inequalities of Brascamp–Lieb type on compact homogeneous spaces of Lie groups. As an application we obtain sharp integral inequalities on the real unit sphere involving functions with some degree of symmetry.

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Invariant Einstein Metrics on Some Homogeneous Spaces of Classical Lie Groups

Canadian Journal of Mathematics ◽

10.4153/cjm-2009-056-2 ◽

2009 ◽

Vol 61 (6) ◽

pp. 1201-1213 ◽

Cited By ~ 12

Author(s):

Andreas Arvanitoyeorgos ◽

V. V. Dzhepko ◽

Yu. G. Nikonorov

Keyword(s):

Riemannian Manifold ◽

Positive Integer ◽

Lie Groups ◽

Einstein Metrics ◽

Homogeneous Spaces ◽

Classical Groups ◽

Stiefel Manifolds ◽

Additional Symmetries

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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Orbit equivalence and rigidity of ergodic actions of Lie groups

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700009251 ◽

1981 ◽

Vol 1 (2) ◽

pp. 237-253 ◽

Cited By ~ 9

Author(s):

Robert J. Zimmer

Keyword(s):

Invariant Measure ◽

Euclidean Space ◽

Lie Groups ◽

Homogeneous Spaces ◽

The Other ◽

Rigidity Theorem ◽

Simple Groups ◽

Orbit Equivalence ◽

Finite Invariant Measure ◽

Lattice Subgroups

AbstractThe rigidity theorem for ergodic actions of semi-simple groups and their lattice subgroups provides results concerning orbit equivalence of the actions of these groups with finite invariant measure. The main point of this paper is to extend the rigidity theorem on one hand to actions of general Lie groups with finite invariant measure, and on the other to actions of lattices on homogeneous spaces of the ambient connected group possibly without invariant measure. For example, this enables us to deduce non-orbit equivalence results for the actions of SL (n, ℤ) on projective space, Euclidean space, and general flag and Grassman varieties.

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