TITLE: A Riemannian metric which doesn't vary smoothly
QUESTION [5 upvotes]: A Riemannian metric $ds^2$ on a differential manifold $M$ of dimension $n\ge 2$ is written as $ds^2=\sum_{i=1}^n\sum_{j=1}^ng_{ij}(x_1,\dots,x_n)dx_idx_j$ where the functions $g_{ij}:M\rightarrow \mathbb{R}$ are twice continuously differentiable functions. 
The derivatives of the $g_{ij}$ are fundamental to define tools as Christoffel symbols, connection, curvature..
My question is: what happens if the $g_{ij}$ still admit derivatives of first and second order in every point of $M$, but these derivatives are not continuous? Is it still possible to define and use Christoffel symbols and curvature tensor (and other tool involving derivatives of $g_{ij}$) or should one expect something going wrong?

REPLY [2 votes]: Beyond the theory of spaces of curvature bounded above/below in the sense of Alexandrov, people also studied scalar curvature defined in the distributional sense. If you are interested in the 2-dimensional case, then the scalar curvature is all what you need: It becomes a (signed) measure on the surface. Take a look at, say, this paper to see how this is done:
Dan A. Lee, Philippe G. LeFloch, The positive mass theorem for manifolds with distributional curvature. 
In fact, they make even weaker regularity assumption than $C^1$: $g\in C^0\cap W_{loc}^{1,n}$ (where $M$ is $n$-dimensional).