May $M$ be a smooth manifold with boundary $\partial M$. Metrics and Connection can be defined everywhere. But now this manifold is cut by a smooth hypersurface $A$ and the cut goes along $M \cap A$ such that it is subdivided into two manifolds $M_1,M_2$ with $M_1 \cup M_2 = M-A$ (disjoint union is used).
Suppose that there is a closed interval $I = [0,1]$ such that a "process" $\phi (M \times I)$ with $\phi(x,0) = x$, $\phi(x,1) = z$ and $x \in M, z \in M_1 \cup M_2$ can be defined. Clearly the manifold is smooth on $M_1 \cup M_2$ but what is on the boundaries $\partial M_1 \cap A, \partial M_2 \cap A$ where a cut takes place? One observes that the normal field $n$ on $\partial M_1 \cap A, \partial M_2 \cap A$ is discontinuous which implies that the curvature and torsion is not defined there. The map $\phi(x,t)$ is also non-differentiable in Argument $t$.
How can I describe the geometry on the cut Surface? I know how to describe surfaces e.g. with Gauss curvature but how do I deal with the problem given above? What area of geometry deals with such problems (It isn't Riemannian geometry because of the discontinuities, which geometry is it then?)? Every hint would be greatly appreciated?