Lower bounds for natural pseudodistances via size functions

Let us consider two $C^1$ closed homeomorphic manifolds $mathcal{M}$, $mathcal{N}$ and two $C^1$ functions $varphi:{mathcal{M}}rightarrow mathbb{R}$, $psi:mathcal{N}rightarrow mathbb{R}$, called measuring functions. 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 show that size functions allow us to get a lower bound for $d$. Furthermore, we prove that this lower bound can be assumed equal either to $|c'-c''|$ or to $frac{1}{2}|c'-c''|$, where $c'$, $c''$ are two suitable critical values of the measuring functions.