Every Proper Smooth Action of a Lie Group is Equivalent to a Real Analytic Action: A Contribution to Hilbert’s Fifth Problem

1996 ◽  
pp. 189-220
Author(s):  
Sören Illman
Keyword(s):  
Lie Group ◽  
Smooth Action ◽  
Real Analytic ◽  
Hilbert’S Fifth Problem ◽  
Analytic Action

The Topology of a Group Action

2020 ◽  
pp. 205-212
Author(s):  
Loring W. Tu
Keyword(s):  
Fixed Point ◽  
Vector Bundle ◽  
Group Action ◽  
Lie Group ◽  
Tubular Neighborhood ◽  
Compact Lie Group ◽  
Fixed Point Set ◽  
Smooth Action ◽  
Point Set ◽  
Neighborhood Theorem

This chapter describes the topology of a group action. It proves some topological facts about the fixed point set and the stabilizers of a continuous or a smooth action. The chapter also introduces the equivariant tubular neighborhood theorem and the equivariant Mayer–Vietoris sequence. A tubular neighborhood of a submanifold S in a manifold M is a neighborhood that has the structure of a vector bundle over S. Because the total space of a vector bundle has the same homotopy type as the base space, in calculating cohomology one may replace a submanifold by a tubular neighborhood. The tubular neighborhood theorem guarantees the existence of a tubular neighborhood for a compact regular submanifold. The Mayer–Vietoris sequence is a powerful tool for calculating the cohomology of a union of two open subsets. Both the tubular neighborhood theorem and the Mayer–Vietoris sequence have equivariant counterparts for a G-manifold where G is a compact Lie group.

Rigidity of Hamiltonian Actions

2003 ◽  
Vol 46 (2) ◽  
pp. 277-290 ◽  
Author(s):  
Frédéric Rochon
Keyword(s):  
General Theory ◽  
Lie Group ◽  
Geometric Analysis ◽  
Smooth Action ◽  
Symplectic Action ◽  
Hamiltonian Actions

AbstractThis paper studies the following question: Given an ω′-symplectic action of a Lie group on a manifoldMwhich coincides, as a smooth action, with a Hamiltonian ω-action, when is this action a Hamiltonian ω′-action? Using a result of Morse-Bott theory presented in Section 2, we show in Section 3 of this paper that such an action is in fact a Hamiltonian ω′-action, provided thatM is compact and that the Lie group is compact and connected. This result was first proved by Lalonde-McDuff-Polterovich in 1999 as a consequence of a more general theory that made use of hard geometric analysis. In this paper, we prove it using classical methods only.

Non-abelian cohomology of abelian Anosov actions

2000 ◽  
Vol 20 (1) ◽  
pp. 259-288 ◽  
Author(s):  
ANATOLE KATOK ◽  
VIOREL NIŢICĂ ◽  
ANDREI TÖRÖK
Keyword(s):  
Discrete Time ◽  
Lie Group ◽  
Global Information ◽  
Infinite Dimensional ◽  
Anosov Actions ◽  
Geometric Character ◽  
Higher Rank ◽  
Real Analytic ◽  
A New Technique ◽  
Discrete Time Case

We develop a new technique for calculating the first cohomology of certain classes of actions of higher-rank abelian groups (${\mathbb Z}^k$ and ${\mathbb R}^k$, $k\ge 2$) with values in a linear Lie group. In this paper we consider the discrete-time case. Our results apply to cocycles of different regularity, from Hölder to smooth and real-analytic. The main conclusion is that the corresponding cohomology trivializes, i.e. that any cocycle from a given class is cohomologous to a constant cocycle. The principal novel feature of our method is its geometric character; no global information about the action based on harmonic analysis is used. The method can be developed to apply to cocycles with values in certain infinite dimensional groups and to rigidity problems.

scholarly journals Lie theory for asymptotic symmetries in general relativity: The BMS Group

2022 ◽  
Author(s):  
David Nicolas Prinz ◽  
Alexander Schmeding
Keyword(s):  
General Relativity ◽  
Lie Group ◽  
Group Structure ◽  
Symmetry Groups ◽  
Asymptotic Symmetry ◽  
Infinite Dimensional ◽  
Real Analytic ◽  
Definition Of ◽  
Future Work ◽  
Asymptotic Symmetries

Abstract We study the Lie group structure of asymptotic symmetry groups in General Relativity from the viewpoint of infinite-dimensional geometry. To this end, we review the geometric definition of asymptotic simplicity and emptiness due to Penrose and the coordinate-wise definition of asymptotic flatness due to Bondi et al. Then we construct the Lie group structure of the Bondi--Metzner--Sachs (BMS) group and discuss its Lie theoretic properties. We find that the BMS group is regular in the sense of Milnor, but not real analytic. This motivates us to conjecture that it is not locally exponential. Finally, we verify the Trotter property as well as the commutator property. As an outlook, we comment on the situation of related asymptotic symmetry groups. In particular, the much more involved situation of the Newman--Unti group is highlighted, which will be studied in future work.

scholarly journals Analytic Jet Parametrization for CR Automorphisms of Some Essentially Finite CR Manifolds

2008 ◽  
Vol 189 ◽  
pp. 155-168
Author(s):  
Sung-Yeon Kim
Keyword(s):  
Uniform Convergence ◽  
Lie Group ◽  
Group Structure ◽  
Cr Manifolds ◽  
Real Analytic ◽  
The Stability

AbstractIn this paper we construct analytic jet parametrizations for the germs of real analytic CR automorphisms of some essentially finite CR manifolds on their finite jet at a point. As an application we show that the stability groups of such CR manifolds have Lie group structure under composition with the topology induced by uniform convergence on compacta.

scholarly journals Extension theorems for smooth functions on real analytic spaces and quotients by Lie groups and smooth stability

1977 ◽  
Vol 24 (4) ◽  
pp. 440-457
Author(s):  
G. S. Wells
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Analytic Space ◽  
Main Step ◽  
Compact Lie Group ◽  
Orbit Type ◽  
Smooth Functions ◽  
Extension Theorems ◽  
Real Analytic

AbstractExtension theorems are proved for smooth functions on a coherent real analytic space for which local defining functions exist which are finitely determined in the sense of J. Mather, (1968), and for smooth functions invariant under the action of a compact lie groupG. thus providing the main step in the proof that smooth infinitesimal stability implies smooth stability in the appropriate categories. In addition the remaining step for the category ofCxG-manifolds of finite orbit type is filled in.

scholarly journals The Lie group of real analytic diffeomorphisms is not real analytic

2015 ◽  
pp. 1-32 ◽  
Author(s):  
Rafael Dahmen ◽  
Alexander Schmeding
Keyword(s):  
Lie Group ◽  
Real Analytic

scholarly journals The Lie Group in Infinite Dimension

2011 ◽  
Vol 2011 ◽  
pp. 1-35 ◽  
Author(s):  
V. Tryhuk ◽  
V. Chrastinová ◽  
O. Dlouhý
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Vector Fields ◽  
Dimensional Space ◽  
Finite Dimensional Space ◽  
Infinite Dimensional ◽  
Symmetries Of Differential Equations ◽  
Smooth Action ◽  
Finite Dimensional ◽  
Infinitesimal Transformations

A Lie group acting on finite-dimensional space is generated by its infinitesimal transformations and conversely, any Lie algebra of vector fields in finite dimension generates a Lie group (the first fundamental theorem). This classical result is adjusted for the infinite-dimensional case. We prove that the (local,C∞smooth) action of a Lie group on infinite-dimensional space (a manifold modelled onℝ∞) may be regarded as a limit of finite-dimensional approximations and the corresponding Lie algebra of vector fields may be characterized by certain finiteness requirements. The result is applied to the theory of generalized (or higher-order) infinitesimal symmetries of differential equations.

A Property of Lie Group Orbits

2000 ◽  
Vol 43 (1) ◽  
pp. 47-50
Author(s):  
Mladen Božičević
Keyword(s):  
Lie Group ◽  
Analytic Manifold ◽  
Conormal Variety ◽  
Real Analytic Manifold ◽  
Real Analytic ◽  
Equivariant Sheaf

AbstractLet G be a real Lie group and X a real analytic manifold. Suppose that G acts analytically on X with finitely many orbits. Then the orbits are subanalytic in X. As a consequence we show that the micro-support of a G-equivariant sheaf on X is contained in the conormal variety of the G-action.

Sign in / Sign up

Export Citation Format

Share Document