Recently I began to consider algebraic surfaces, that is, the zero set of a polynomial in 3 (or more variables). My algebraic geometry background is poor, and I'm more used to differential and Riemannian geometry. Therefore, I'm looking for the relations between the two areas. I should also mention, that I'm interested in the realm of real surfaces, i.e. subsets of $\mathbb{R}^n$.

On my desk you could find the following books: **Algebraic Geometry** by *Hartshorne*, **Ideals, Varieties, and Algorithms** by *Cox & Little & O'Shea*, **Algorithms in Real Algebraic Geometry** by *Basu & Pollack & Roy* and **A SINGULAR Introduction to Commutative Algebra** by *Greuel & Pfister*. Unfortunately, neither of them introduced notions and ideas I'm looking for.

If I get it right, please correct me if I'm wrong, locally, around non-singular points, an algebraic surface behaves very nicely, for example, it is smooth. Here's the first question: *is it locally (about non-singular point) a smooth manifold? Is it a Riemannian manifold, having, for instance, the metric induced from the Euclidean space?*

Further questions I have are, for example:

- Can I define
*geodesics*(either in the sense of length minimizer or straight curves) in the non-singular areas of the surface? Can they pass singularities? - How about
*curvature*? Is it defined for these objects? - Can we talk about
*convexity*of subsets of the algebraic surface? - What other tools and term can be imported from differential/Riemannian geometry?

I will be grateful for any hint, tip and lead in the form of either answers to my questions, or references to books/papers which can be helpful, or any other sort of help.