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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.