  We consider upper bounds on the growth of $L^p$ norms of restrictions of
eigenfunctions and quasimodes to geodesic segments in a nonpositively curved
manifold in the high frequency limit. This sharpens results of Chen and Sogge
as well as Xi and Zhang, which showed that the crux of the problem is to
establish bounds on the mixed partials of the distance function on the covering
manifold restricted to geodesic segments. The innovation in this work is the
development of a formula for the third variation of arc length on the covering
manifold, which allows for a coordinate free expressions of these mixed
partials.
