Lifting differentiable curves from orbit spaces

TL;DR

This paper establishes conditions under which differentiable curves in orbit spaces of compact Lie group representations can be lifted to Lipschitz continuous curves in the original space, providing explicit bounds and smoothness results.

Contribution

It proves that certain differentiable curves in orbit spaces can be lifted to Lipschitz curves in the original space with explicit bounds, and that smooth curves admit smooth lifts, extending previous results.

Findings

Lipschitz lifts exist for $C^{d-1,1}$-curves in the orbit space.

$C^d$-curves in the orbit space have $C^1$-lifts.

Multivariable extensions for finite groups are obtained.

Abstract

Let $\rho : G \rightarrow \operatorname{O}(V)$ be a real finite dimensional orthogonal representation of a compact Lie group, let $\sigma = (\sigma_1,\ldots,\sigma_n) : V \to \mathbb R^n$, where $\sigma_1,\ldots,\sigma_n$ form a minimal system of homogeneous generators of the $G$-invariant polynomials on $V$, and set $d = \max_i \operatorname{deg} \sigma_i$. 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.