In this lecture of Yau's on the Existence of complete Kähler-Einstein metrics with negative scalar curvature he mentions the following, I quote:
Negative holomorphic sectional curvature is a rather interesting statement. You don't need [a] smooth[ness] assumption on the metric to describe negative holomorphic sectional curvature. Because if you take any curve, sitting in a manifold, the induced metric will have negative holomorphic sectional curvature. So the smoothness assumption is just that any metric on the ambient manifold induce[s] on the curve, [a metric of] negative holomorphic sectional curvature, let's say equal to $-1$, for example. So you only need [a] smoothness assumption on all the curves. And that is easy to describe. In that form, I'm not sure that this proof [of the Wu--Yau theorem] can be carried out actually, although it would be interesting.
Are examples of such metrics easy to construct? Namely, examples of non-smooth metrics which induce a smooth metric on every curve?
