Lifting differentiable curves from orbit spaces

Abstract

Let $\rho :G\to O(V)$ be a real finite dimensional orthogonal representation of a compact Lie group, let $\sigma =({\sigma }_1,\dots ,{\sigma }_n):V\to {\mathbb{R}}^n$, where ${\sigma }_1,\dots ,{\sigma }_n$ form a minimal system of homogeneous generators of the $G$-invariant polynomials on $V$, and set $d={\max }_i\mathrm{deg}{\sigma }_i$. We prove that for each $C^{d-1,1}$-curve $c$ in $\sigma (V)\subseteq {\mathbb{R}}^n$ there exits a locally Lipschitz lift over $\sigma$, i.e., a locally Lipschitz curve $\overline{c}$ in $V$ so that $c=\sigma \circ \overline{c}$, and we obtain explicit bounds for the Lipschitz constant of $\overline{c}$ in terms of $c$. Moreover, we show that each $C^d$-curve in $\sigma (V)$ admits a $C^1$-lift. For finite groups $G$ we deduce a multivariable version and some further results.

25 pages; section on orbit spaces as differentiable spaces added, some typos corrected; accepted for publication in Transformation Groups

References in corpus (2)

• A new proof of Bronshtein's theorem
• Lifting smooth curves over invariants for representations of compact Lie groups. II

Cited by in corpus (2)

• Arc-smooth functions on closed sets
• Sobolev Lifting over Invariants