On the question of the existence of a canonically defined distance function for metrics with indefinite signature on a given manifold
Let ℜ be the set of all inner products with a given indefinite signature on a finite-dimensional vector spaceV. It is shown that any continuous pseudometricd on ℜ which is “canonical,” i.e., invariant with respect to all automorphisms ofV, is of the formd(g, h)=f(υ g /υ h ), whereυ g , υ h denote the volume elements associated withg andh, respectively, andf is a real-valued function; i.e., there are no nontrivial distance functions. This result shows that a reasonable distance function between metric fields on a manifoldM cannot be obtained in terms of canonical distances of the inner products induced in the tangent spaces. A similar result is proved even for uniform structures instead of pseudometrics.