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Peter Sarnak and coworkers have extensively studies the ergodicity of geodesic flow on negatively curved manifolds but completely integrable systems such as spheres do not have this ergodicity with the standard metric.  Now Zelditch had proved that using the degeneracy of the eigenspaces of the Laplacian on $S^2$ and using the Haar measure on unitary matrices preserving the eigenspaces, there is generic ergodicity almost surely.  We happen to believe that the actual universe is deterministic and a four-sphere where the empirical material universe is an evolving subspace.  So another example of a positively curved manifold with ergodicity in geodesic flow is addressed according a 2014 paper referring to Burns and Gedeon, “Metrics with ergodic geodesic flows on spheres”