The Nash-Tognoli theorem says that every closed, smooth manifold is diffeomorphic to a real algebraic variety.
Suppose I wanted to study the Ricci curvature of some class of manifolds.
Is there a "good reason" to look at manifolds as algebraic varieties?
I.e. is there an example where geometric or differential questions can be answered using algebra (by viewing the manifold as a variety) that cannot be (perhaps as easily) answered through "regular" geometry?