In geometry there is plenty of examples in which the following happens:
- Some elements are considered equivalent, in some topological or algebraic sense
- We take the quotient
- The metric is usually not invariant within the classes,
but rather, on the contrary:
- Each class admits one and only one "optimal" element for the metric.
The most famous examples are:
- De Rham cohomology classes and harmonic forms (Hodge)
- Homotopy classes of loops and geodesics (Cartan - if I'm not mistaken).
In a way, in these constructions, there is a sort of "duality" between the topology and the metric. "Duality" in the sense of, for example, magnetic monopoles in physics. In the case of harmonic forms, the duality is explicit, as there is indeed a dual complex given by the codifferential.
Now the question is:
Is this a universal phenomenon?
Are there counterexamples in which the metric plays a different role (e.g. is well-defined in the class, rather than picking the optimal element)? Are there more, or more general, examples in which the metric picks out a term in a topological class? Is there a formal generalization of this phenomenon (something about how "not functorial" the metric is)? Can we always define "dual classes", as for the Hodge case?
(I hope that it's clear what I'm asking. Feel free to edit question, tags, or anything you find inappropriate.)