- #1
Alesak
- 111
- 0
When reading other threads, following question crept into my mind:
When given a manifold, why shouldn't I give it distance function by giving it a simple metric function, that is MxM→ℝ with the usual axioms? I could happily measure distances in coordinate-independent way for ever after...
Why do I need to use the horrid construction of symmetric positive-definite covariant tensor field of second degree?
Also, what stops me from defining Riemannian metric on topological manifold? In definition of metric space we need no smoothness either.
When given a manifold, why shouldn't I give it distance function by giving it a simple metric function, that is MxM→ℝ with the usual axioms? I could happily measure distances in coordinate-independent way for ever after...
Why do I need to use the horrid construction of symmetric positive-definite covariant tensor field of second degree?
Also, what stops me from defining Riemannian metric on topological manifold? In definition of metric space we need no smoothness either.