Connected components of partition preserving diffeomorphisms

Abstract

Let $f:\mathbb{R}^2\to\mathbb{R}$ be a homogeneous polynomial and $\mathcal{S}(f)$ be the group of diffeomorphisms $h$ of $\mathbb{R}^2$ preserving $f$, i.e. $f \circ h =f$. Denote by $\mathcal{S}_{\mathrm{id}}(f)^{r}$, $(0\leq r \leq \infty)$, the identity component of $\mathcal{S}(f)$ with respect to the weak Whitney $C^{r}_{W}$-topology. We prove that $\mathcal{S}_{\mathrm{id}}(f)^{\infty} = \cdots = \mathcal{S}_{\mathrm{id}}(f)^{1}$ for all $f$ and that $\mathcal{S}_{\mathrm{id}}(f)^{1} ot= \mathcal{S}_{\mathrm{id}}(f)^{0}$ if and only if $f$ is a product of at least two distinct irreducible over $\mathbb{R}$ quadratic forms.