The Weierstrass approximation theorem and a characterization of the unit circle

We study real algebraic morphisms from nonsingular real algebraic varieties $X$ with $\dim X \geq 1$ into nonsingular real algebraic curves $C$. We show, among other things, that the set of real algebraic morphisms from $X$ into $C$ is never dense in the space of all $\mathcal C^\infty$ maps from $X$ into $C$, unless $C$ is biregularly isomorphic to a Zariski open subset of the unit circle.