I hope these are not to vague questions for MO.

Is there an analog of the concept of a Riemannian metric, in algebraic geometry?

Of course, transporting things literally from the differential geometric context, we have to forget about the notion of positive definiteness, cause a bare field has no ordering. So perhaps we're looking to an algebro geometric analog of *semi-* Riemannian geometry.

Suppose to consider a pair $(X,g)$, where $X$ is a (perhaps smooth) variety and $g$ is a nondegenerate section of the second symmetric power of the tangent bundle (or sheaf) of $X$.

What can be said about this structure? Can some results of DG be reproduced in this context? Is there a literature about this things?

differential topologyrely on partitions of unity. - You do have a well developped theory of holomorphic-symplectic (and algebraic-symplectic) manifolds, though. – Qfwfq Mar 25 '10 at 18:33concentratecurvature in a subvariety of lower dimension. Not being an algebraic geometer myself, I can't (alas) produce a clear example of this practice off the top of my head. – some guy on the street Apr 15 '10 at 16:02