This is related to a previous question of mine.
Basically, it has occurred to me that there are two distinct ways in which a $k-$dimensional smooth manifold looks locally like $\mathbb{R}^k$: (1) at each point it has a well-defined tangent space which is isomorphic in many ways to $\mathbb{R}^k$ (with the points of the tangent space, except for the origin, not being points of the manifold itself) (2) at each point it has a neighborhood which is homeomorphic to $\mathbb{R}^k$ (with the points of the neighborhood obviously being points of the manifold itself).
So then does the difference between differentiable manifolds and non-differentiable topological manifolds come down to the following:
Differentiable manifolds resemble $\mathbb{R}^k$ in the sense of both (1) and (2).
Non-differentiable topological manifolds resemble $\mathbb{R}^k$ only in the sense of (2).
And then an intuitive way of thinking of the implicit function/inverse function/rank theorems is that they prove that $(1) \implies (2)$?
Is the existence of exponential maps for Riemannian manifolds an alternative way to show that $(1) \implies (2)$?