Five–Dimensional φ–Symmetric Spaces

It is still an open problem whether Riemannian manifolds all of whose local geodesic symmetries are volume–preserving (i.e., D’Atri spaces) or more generally, ball–homogeneous spaces, and C-spaces are locally homogeneous or not. We provide some partial positive answers by proving that five–dimensional locally φ–symmetric spaces can be characterized as Sasakian spaces which are ball–homogeneous with η-parallel Ricci tensor or D’Atri spaces or C–spaces. We also prove that all K–contact metric manifolds, and hence all Sasakian manifolds, which are harmonic have constant curvature one. Mathematics Subject Classification: 53B20, 53C25, 53C30.