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

#### 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\$.
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P. Donatini; P. Frosini
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11585/12212`
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