Manifolds of absolutely continuous functions with values in an infinite-dimensional manifold and regularity properties of half-Lie groups

Abstract

For $p\in [1,\infty]$, we define a smooth manifold structure on the set of absolutely continuous functions $\gamma\colon [a,b]\to N$ with $L^p$-derivatives for each smooth manifold $N$ modeled on a sequentially complete locally convex topological vector space which admits a local addition. Smoothness of natural mappings between spaces of absolutely continuous functions is discussed. For $1\leq p <\infty$ and $r\in \mathbb{N}$ we show that the right half-Lie groups $\text{Diff}_K(\mathbb{R})$ and $\text{Diff}(M)$ are $L^p$-semiregular. Here $K$ is a compact subset of $\mathbb{R}^n$ and $M$ is a compact smooth manifold. For the preceding examples, the evolution map $\text{Evol}$ is continuous.