Gromov and Thurston proved in "Pinching constants for hyperbolic manifolds" that any finite ramified covering of a compact hyperbolic manifold, along a codimension $2$ totally geodesic submanifold, can be endowed with a riemannian metric of negative sectional curvature. Pansu mentions in a survey (4/TSG\_1985-1986\_*4*\_101\_0/TSG\_1985-1986\_*4*\_101\_0.pdf">http://archive.numdam.org/ARCHIVE/TSG/TSG\_1985-1986\_*4*/TSG\_1985-1986\_*4*\_101\_0/TSG\_1985-1986\_*4*\_101\_0.pdf) that he doesn't know how to make Gromov and Thurston's calculation in the general case. Is it possible to generalize ? 


Namely let $M$ be a compact manifold endowed with a riemannian metric of negative sectional curvature. Suppose $X \longrightarrow M$ is a ramified covering of finite degree, along a compact submanifold of codimension $2$. Can $X$ be endowed with a riemannian metric of negative sectional curvature ? Can a counterexample be found when the locus of ramification is not totally geodesic ? Can one hope a pinching of the curvature like Gromov and Thurston ? Is the covering, in any reasonable sense, "more negatively" than the base ?



Are ramified covering of negatively curved manifolds negatively curved?
Suppose that $M$ is a closed hyperbolic 3-manifold and $K$ is a trivial knot contained in a ball $B\subset M$. Then $M$ admits a 2-fold covering $M'\to M$ ramified over $K$ so that $M'$ is not even aspherical (it is homeomorphic to the connected sum of two copies of $M$). Therefore, $M'$ cannot admit a negatively curved metric. 


Do you want the ramification locus to be totally-geodesic? If the ramification locus is totally geodesic, then the covering space $M'$ has a locally $CAT(-k)$-metric where $k>0$ and the original manifold $M$ has lower curvature bound $-k$, so in this sense, it is more negatively curved than the original manifold $M$. In general, I do not think there is a (known) smoothing procedure for this metric among negatively curved metrics. Note that in the context of nonpositively curved manifolds, a branched covering ramified over a totally geodesic submanifold may not admit a smooth metric of nonpositive curvature (cf. Exercise 1 in Gromov-Ballmann-Schroeder, which was finally solved this year, almost 30 years after the book was published). 


