Is any real smooth manifold diffeomorphic to a real affine algebraic variety? (I.e. is there an "algebraic" Whitney embedding theorem?)
And are all possible ways of realizing a manifold $M$ as an algebraic variety equivalent? I.e. suppose $M$ is diffeomorphic to varieties $V_1$ and $V_2$, are these isomorphic in the algebraic category?
Admittely I'm just asking out of curiosity after reading this question: Can manifolds be uniformly approximated by varieties?