Natural pseudodistance

In size theory, the natural pseudodistance between two size pairs (M,\varphi:M\to \mathbb{R})\ , (N,\psi:N\to \mathbb{R})\ is the value \inf_h \|\varphi-\psi\circ h\|_\infty\ , where h\ varies in the set of all homeomorphisms from the manifold M\ to the manifold N\ and \|\cdot\|_\infty\ is the supremum norm. If M\ and N\ are not homeomorphic, then the natural pseudodistance is defined to be \infty\ . It is usually assumed that M\ , N\ are C^1\ closed manifolds and the measuring functions \varphi,\psi\ are C^1\ . Put another way, the natural pseudodistance measures the infimum of the change of the measuring function induced by the homeomorphisms from M\ to N\ .

The concept of natural pseudodistance can be easily extended to size pairs where the measuring function \varphi\ takes values in \mathbb{R}^m\ .[1]

Main properties

It can be proved [2] that the natural pseudodistance always equals the Euclidean distance between two critical values of the measuring functions (possibly, of the same measuring function) divided by a suitable positive integer k\ . If M\ and N\ are surfaces, the number k\ can be assumed to be 1\ , 2\ or 3\ .[3] If M\ and N\ are curves, the number k\ can be assumed to be 1\ or 2\ .[4] If an optimal homeomorphism \bar h\ exists (i.e., \|\varphi-\psi\circ \bar h\|_\infty=\inf_h \|\varphi-\psi\circ h\|_\infty\ ), then k\ can be assumed to be 1\ .[2]

See also

References

  1. ↑ Patrizio Frosini, Michele Mulazzani, Size homotopy groups for computation of natural size distances, Bulletin of the Belgian Mathematical Society, 6:455-464, 1999.
  2. 1 2 Pietro Donatini, Patrizio Frosini, Natural pseudodistances between closed manifolds, Forum Mathematicum, 16(5):695-715, 2004.
  3. ↑ Pietro Donatini, Patrizio Frosini, Natural pseudodistances between closed surfaces, Journal of the European Mathematical Society, 9(2):231–253, 2007.
  4. ↑ Pietro Donatini, Patrizio Frosini, Natural pseudodistances between closed curves, Forum Mathematicum, 21(6):981–999, 2009.
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