
4:16 PM (0 minutes ago) 

Ladies and Gentlemen,
Although the round four sphere itself has totally integrable geodesic flow, generic hypersurfaces of the sphere have positive topological entropy and therefore ergodic geodesic flow. This is a relatively easy corollary of a theorem that says that for compact manifolds an open dense set of generic metrics have positive topological entropy and therefore ergodic geodesic flow, so some analysis shows that generic embedding metrics into a four sphere (which should be for radius 1/h) which can be characterized by additional GaussCodazzi relations also have ergodic geodesic flow and should therefore fall under the quantum ergodicity theorems. The attached paper has the generic metrics result.