Derived geometry explains how to remove the transversality condition and make sense out of a nontransversal intersection.
For example, if $X$ and $Y$ are embedded submanifolds of a manifold (or spaceform) $Z$, then the intersection of $X$ and $Y$ is a derived manifold of dimension $\dim X+\dim Y-\dim Z$, which can be negative.


For a derived manifold of dimension $-2$, how I can write its metric?
Can someone give me an example of such a metric? 



Metrics on derived smooth manifolds
As far as I am aware, there is nothing in the literature that
treats Riemannian or pseudo-Riemannian metrics on derived smooth manifolds.
However, there is an extensive treatment of symplectic structures
on derived stacks by Pantev, Toën, Vaquié, and Vezzosi:
[Shifted symplectic structures](https://arxiv.org/abs/1111.3209).


Due to the Koszul sign rule, a pseudosymplectic structure (i.e., a symplectic structure without the integrability condition) is nothing else
than a pseudo-Riemannian metric on the cotangent complex shifted in degree by 1.


