Connected components of partition preserving diffeomorphisms

Abstract

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

22 pages, 6 figures. In the previous version only polynomials without multiple factors were considered. Now the result is proved for all homogeneous polynomials. Moreover some proofs are rewritten with more details

References in corpus (1)

• Stabilizers and orbits of smooth functions

Cited by in corpus (1)

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