I am asking because I am wondering if studying smooth manifolds is the same as studying zero loci of smooth functions with non-vanishing gradient.
Let me be a little formal for clarity:
Let $M$ be a smooth manifold, and $i : M \to R^n$ a smooth embedding into $n$-dimensional Euclidean space. Define a function $d: R^n \to R$ by $d(x) = \inf_{m \in M} d(x,m)$. Is the function $d$ smooth?
(What about if compactness is assumed? What happens if the manifold is merely immersed?)
(Edit: I did later realize that the right way to approach my underlying question for an embedding is to use the local normal form, the fact that planes are smooth varieties, and then a partition of unity to patch everything together.)