The isometry group of a compact Lorentz manifold. I

Abstract It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.

Download Full-text

The isometry group of Outer Space

Advances in Mathematics ◽

10.1016/j.aim.2012.07.011 ◽

2012 ◽

Vol 231 (3-4) ◽

pp. 1940-1973 ◽

Cited By ~ 6

Author(s):

Stefano Francaviglia ◽

Armando Martino

Keyword(s):

Isometry Group ◽

Outer Space

Download Full-text

An S2 $times$ S1 bundle over CP2 - a solution of d = 11 supergravity with isometry group (SU(3)/Z3) $times$ U(2)

Classical and Quantum Gravity ◽

10.1088/0264-9381/3/2/519 ◽

1986 ◽

Vol 3 (2) ◽

pp. 261-261

Author(s):

B Dolan

Keyword(s):

Isometry Group

Download Full-text

Conformal geodesics on gravitational instantons

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004121000463 ◽

2021 ◽

pp. 1-32

Author(s):

MACIEJ DUNAJSKI ◽

PAUL TOD

Keyword(s):

Magnetic Field ◽

Geodesic Flow ◽

Isometry Group ◽

Three Dimensional ◽

First Integrals ◽

Complete Integrability ◽

Conformal Killing ◽

Gravitational Instantons ◽

Geodesic Equations ◽

Jacobi Equations

Abstract We study the integrability of the conformal geodesic flow (also known as the conformal circle flow) on the SO(3)–invariant gravitational instantons. On a hyper–Kähler four–manifold the conformal geodesic equations reduce to geodesic equations of a charged particle moving in a constant self–dual magnetic field. In the case of the anti–self–dual Taub NUT instanton we integrate these equations completely by separating the Hamilton–Jacobi equations, and finding a commuting set of first integrals. This gives the first example of an integrable conformal geodesic flow on a four–manifold which is not a symmetric space. In the case of the Eguchi–Hanson we find all conformal geodesics which lie on the three–dimensional orbits of the isometry group. In the non–hyper–Kähler case of the Fubini–Study metric on $\mathbb{CP}^2$ we use the first integrals arising from the conformal Killing–Yano tensors to recover the known complete integrability of conformal geodesics.

Download Full-text