If $g$ is a smooth riemannian metric on $M$ with nonpositive sectional curvature, the Rauch comparison theorem implies that $(M,d_g)$ is a negatively curved metric space (every point has a geodesically convex neighborhood in which triangle satisfy the $\bf{CAT}(0)$-inequality). I would like to know if a $C^{1,1}$ version of that statement is true, namely :
If $g$ is a $C^{1,1}$ metric on $M$ with almost everywhere sectional curvature satisfying $K\leq 0$, does it hold that $(M,d_g)$ is a negatively curved metric space ?
This should be written somewhere, so a reference would be appreciated.
