New characterization of \(\phi\)-symmetric spaces
Sasakian \(\phi\)-symmetric spaces have been introduced and studied by \textit{T. Takahashi} in [Tôhoku Math. J., II. Ser. 29, 91-113 (1977; Zbl 0343.53030)] and the notion of a \(\phi\)-geodesic symmetry, used to define them, has been explored further in [the authors, ibid. 39, 373-383 (1987; Zbl 0632.53039)]. In this note we give three characterizations of Sasakian \(\phi\)-symmetric spaces. The first is technical in nature and is analogous to the fact that a Kähler manifold is locally symmetric if and only if \((\nabla_ XR)_{XJXXJX}=0\) for all X. The second is geometric in nature and is analogous to the following characteristic property of symmetric spaces. Let m be the midpoint of a sufficiently short geodesic segment joining p and q. Then the geodesic spheres of radius equal to the distance d(m,p) centered at p and q have the same shape operator at m. Finally, we have previously observed [loc. cit.] that a complete, simply connected locally \(\phi\)-symmetric space is a naturally reductive homogeneous space. We show conversely that a naturally reductive homogeneous space with an invariant Sasakian structure is \(\phi\)- symmetric.
scientific article; zbMATH DE number 4096310