Focussing on the mathematics related to the recent proof of ergodicity of the (Weil-Petersson)
geodesic flow on a nonpositively curved space whose points are negatively curved metrics on
surfaces this book provides a broad introduction to an important current area of research. It
offers original textbook-level material suitable for introductory or advanced courses as well
as deep insights into the state of the art of the field making it useful as a reference and
for self-study. The first chapters introduce hyperbolic dynamics ergodic theory and geodesic
and horocycle flows and include an English translation of Hadamard's original proof of the
Stable-Manifold Theorem. An outline of the strategy motivation and context behind the
ergodicity proof is followed by a careful exposition of it (using the Hopf argument) and of the
pertinent context of Teichmüller theory. Finally some complementary lectures describe the deep
connections between geodesic flows in negative curvature and Diophantine approximation.
