Geometry of tubes about characteristic curves on Sasakian manifolds (Q1197541)

Let \(M\) be a Sasakian manifold with contact form \(\eta\), characteristic vector field \(\xi\) and fundamental collineation \(\varphi\). If the sectional curvatures of plane sections spanned by a vector \(X\) orthogonal to \(\xi\) and \(\varphi X\) are constant, \(M\) is called a Sasakian space form (or said to be of constant \(\varphi\)-sectional curvature). If \(\varphi^ 2(\nabla_ VR)_{XY}Z=0\) for \(V\), \(X\), \(Y\), \(Z\) orthogonal to \(\xi\), \(M\) is said to be locally \(\varphi\)-symmetric, [\textit{J. Takahashi}, Tohoku Math. J., II. Ser. 29, 91-113 (1977; Zbl 0343.53030)]. Let \(\sigma\) be an integral curve of \(\xi\) and \(P_ r\) the geodesic tube about \(\sigma\) of radius \(r\). Finally for \(p \in P_ r\), put \(p=\exp_ m(ru)\), \(m \in \sigma\); where \(u\) is a unit normal to \(\sigma\) at \(m\) and denote by \(\kappa_ \sigma\) the curvature at \(p\) of the geodesic of \(P_ r\) tangent to \(\varphi u\). The authors prove that a Sasakian manifold of dimension \(\geq 5\) is a Sasakian space form if and only if the integral curves of \(\varphi u\) on each \(P_ r\) are geodesics if and only if \(\varphi u\) is a Killing field on each \(P_ r\). It is also shown that a Sasakian manifold of dimension \(\geq 5\) is locally \(\varphi\)-symmetric if and only if for all \(u\) (as above) and all small \(r\), \(\kappa_ \sigma(\exp_ \sigma(ru))=\kappa_ \sigma(\exp_ \sigma(-ru))\). Similar characterizations are also obtained using the Ricci operator of the geodesic tubes instead of the shape operator. Proofs make use of the fact that \(\xi\) induces a local fibration over a Kähler manifold \(B\) and that \(M\) is a Sasakian space form if and only if \(B\) is a complex space form, respectively that \(M\) is locally \(\varphi\)-symmetric if and only if \(B\) is locally Hermitian symmetric, one of the main results of the Takahashi paper referred to above.