- #1
RockyMarciano
- 588
- 43
I was wondering about the following scenario, we have a certain differentiable manifold with the standard topology not induced by any previous metric structure on the manifold. There is no natural way to identify a vector with its dual(no canonical isomorphism between them),
If we had to define the length of a curve and we were not allowed to use an induced metric from a prespecified metric structure, would it be necessary to make a choice of inner product for each of the uncountable points in the curve and therefore would we have to appeal to the axiom of choice?
If we had to define the length of a curve and we were not allowed to use an induced metric from a prespecified metric structure, would it be necessary to make a choice of inner product for each of the uncountable points in the curve and therefore would we have to appeal to the axiom of choice?