The Regularity Problem for Lie Groups with Asymptotic Estimate Lie Algebras

We solve the regularity problem for Milnor's infinite dimensional Lie groups in the asymptotic estimate context. Specifically, let $G$ be a Lie group with asymptotic estimate Lie algebra $\mathfrak{g}$, and denote its evolution map by $\mathrm{evol}\colon \mathrm{D}\equiv \mathrm{dom}[\mathrm{evol}]\rightarrow G$, i.e., $\mathrm{D}\subseteq C^0([0,1],\mathfrak{g})$. We show that $\mathrm{evol}$ is $C^\infty$-continuous on $\mathrm{D}\cap C^\infty([0,1],\mathfrak{g})$ if and only if $\mathrm{evol}$ is $C^0$-continuous on $\mathrm{D}\cap C^0([0,1],\mathfrak{g})$. We furthermore show that $G$ is k-confined for $k\in \mathbb{N}\sqcup\{\mathrm{lip},\infty\}$ if $G$ is constricted. (The latter condition is slightly less restrictive than to be asymptotic estimate.) Results obtained in a previous paper then imply that an asymptotic estimate Lie group $G$ is $C^\infty$-regular if and only if it is Mackey complete, locally $\mu$-convex, and has Mackey complete Lie algebra - In this case, $G$ is $C^k$-regular for each $k\in \mathbb{N}_{\geq 1}\sqcup\{\mathrm{lip},\infty\}$ (with ``smoothness restrictions'' for $k\equiv\mathrm{lip}$), as well as $C^0$-regular if $G$ is even sequentially complete with integral complete Lie algebra.Comment: 27 pages. Version as published at Indagationes Mathematicae (title refined; presentation improved; proof of Lemma 9 revised

A Characterization of Invariant Connections

Given a principal fibre bundle with structure group $S$, and a fibre transitive Lie group $G$ of automorphisms thereon, Wang's theorem identifies the invariant connections with certain linear maps $\psi\colon \mathfrak{g}\rightarrow \mathfrak{s}$. In the present paper, we prove an extension of this theorem which applies to the general situation where $G$ acts non-transitively on the base manifold. We consider several special cases of the general theorem, including the result of Harnad, Shnider and Vinet which applies to the situation where $G$ admits only one orbit type. Along the way, we give applications to loop quantum gravity

Regularity of Lie Groups

We solve the regularity problem for Milnor's infinite dimensional Lie groups in the $C^0$-topological context, and provide necessary and sufficient regularity conditions for the standard setting ($C^k$-topology). We prove that the evolution map is $C^0$-continuous on its domain $\textit{iff}\hspace{1pt}$ the Lie group $G$ is locally $\mu$-convex. We furthermore show that if the evolution map is defined on all smooth curves, then $G$ is Mackey complete - This is a completeness condition formulated in terms of the Lie group operations that generalizes Mackey completeness as defined for locally convex vector spaces. Then, under the presumption that $G$ is locally $\mu$-convex, we show that each $C^k$-curve, for $k\in \mathbb{N}_{\geq 1}\sqcup\{\mathrm{lip},\infty\}$, is integrable (contained in the domain of the evolution map) $\textit{iff}\hspace{1pt}$ $G$ is Mackey complete and $\mathrm{k}$-confined. The latter condition states that each $C^k$-curve in the Lie algebra $\mathfrak{g}$ of $G$ can be uniformly approximated by a special type of sequence consisting of piecewise integrable curves - A similar result is proven for the case $k\equiv 0$; and we provide several mild conditions that ensure that $G$ is $\mathrm{k}$-confined for each $k\in \mathbb{N}\sqcup\{\mathrm{lip},\infty\}$. We finally discuss the differentiation of parameter-dependent integrals in the standard topological context. In particular, we show that if the evolution map is well defined and continuous on $C^k([0,1],\mathfrak{g})$ for $k\in \mathbb{N}\sqcup\{\infty\}$, then it is smooth thereon $\textit{iff}\hspace{1pt}$ $\mathfrak{g}$ is $\hspace{0.2pt}$ Mackey complete for $k\in \mathbb{N}_{\geq 1}\sqcup\{\infty\}$ $\hspace{1pt}/\hspace{1pt}$ integral complete for $k\equiv 0$. This result is obtained by calculating the directional derivatives explicitly - recovering the standard formulas that hold in the Banach case.Comment: 72 pages. Revised version: notations simplified; oversights corrected; example added to Sect. 3.5.2; Lemma 40 (now Lemma 13) shifted into Sect. 3.5.4; Lipschitz case added to Lemma 23 (now Lemma 24); proof of Lemma 25 (now Lemma 26) revised; Proposition 6 correcte

Decompositions of Analytic 1-Manifolds

In a previous paper, connected analytic 1-dimensional submanifolds with boundary have been classified w.r.t. their symmetry under a given regular Lie group action on an analytic manifold. It was shown that each such submanifold is either free or analytically diffeomorphic to the unit circle or some interval via the exponential map. In this paper, we show that each free connected analytic 1-submanifold naturally splits into symmetry free segments, mutually and uniquely related by the group action. This is proven under the assumption that the action is non-contractive, which is even less restrictive than regularity.Comment: 25 page

Symmetries of Analytic Curves

Analytic curves are classified w.r.t. their symmetries under a regular Lie group action on an analytic manifold. We show that an analytic curve is either exponential or splits into countably many analytic immersive curves; each of them decomposing naturally into symmetry free subcurves mutually and uniquely related by the group action. We conclude that a connected analytic 1-dimensional submanifold is either analytically diffeomorphic to the unit circle or some interval, or that each point (except for at most countably many) admits a symmetry free chart.Comment: 49 page