Let us consider two closed homeomorphic manifolds $mathcal{M}$, $mathcal{N}$ of class $C^1$ and two functions $varphi:{mathcal{M}}rightarrow mathbb{R}$, $psi:mathcal{N}rightarrow mathbb{R}$ of class $C^1$. The natural pseudodistance ${d}$ between the pairs $({mathcal{M}},varphi)$, $({mathcal{N}},psi)$ is defined as the infimum of $Theta(f)stackrel{def}{=}max_{Pin mathcal{M}}|varphi(P)-psi(f(P))|$, as $f$ varies in the set of all homeomorphisms from $mathcal{M}$ onto $mathcal{N}$. In this paper we prove that a suitable multiple of ${d}$ by a positive integer $k$ coincides with the distance between two critical values of the functions $varphi,psi$.
Natural pseudodistances between closed manifolds / P. Donatini; P. Frosini. - In: FORUM MATHEMATICUM. - ISSN 0933-7741. - STAMPA. - 16(5):(2004), pp. 695-715. [10.1515/form.2004.032]
Natural pseudodistances between closed manifolds
DONATINI, PIETRO;FROSINI, PATRIZIO
2004
Abstract
Let us consider two closed homeomorphic manifolds $mathcal{M}$, $mathcal{N}$ of class $C^1$ and two functions $varphi:{mathcal{M}}rightarrow mathbb{R}$, $psi:mathcal{N}rightarrow mathbb{R}$ of class $C^1$. The natural pseudodistance ${d}$ between the pairs $({mathcal{M}},varphi)$, $({mathcal{N}},psi)$ is defined as the infimum of $Theta(f)stackrel{def}{=}max_{Pin mathcal{M}}|varphi(P)-psi(f(P))|$, as $f$ varies in the set of all homeomorphisms from $mathcal{M}$ onto $mathcal{N}$. In this paper we prove that a suitable multiple of ${d}$ by a positive integer $k$ coincides with the distance between two critical values of the functions $varphi,psi$.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.