D'Atri spaces and \(C\)-spaces in flow geometry (Q1265332)

D'Atri spaces are Riemannian manifolds all of whose local geodesic symmetries are volume-preserving. \(C\)-spaces are defined as Riemannian manifolds such that the Jacobi operators have constant eigenvalues along the corresponding geodesics. It is still an interesting open problem if D'Atri spaces and \(C\)-spaces are locally homogeneous. In the present paper, the positive answer is given for dimensions not greater than five under the condition that the manifold is endowed with a normal or normal contact Riemannian flow generated by a unit Killing vector field. The positive answer is obtained through explicit classifications.