Volume-preserving geodesic symmetries on four-dimensional 2-Stein spaces (Q1819380)

It is an open question whether a Riemannian manifold whose local geodesic symmetries are all volume-preserving is necessarily locally homogeneous. The authors prove this in the case of four-dimensional 2-Stein spaces, the assertion being that the manifold is flat or locally isometric to a rank one symmetric space.