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?



Algebraic (semi-) Riemannian geometry ?
Joel Kamnitzer had a very similar question a couple years ago, that prompted a nice discussion at the [Secret Blogging Seminar](http://sbseminar.wordpress.com/2007/11/15/algebraic-riemannian-manifolds/). I'm afraid no one ended up citing any literature, and I have been unable to find anything with a quick Google search, but that doesn't rule out the possibility of existence.


