Characterizations of complex space forms and locally Hermitian symmetric spaces by geodesic tubes (Q1911823)

Let \((M,g,J)\) be a Kähler manifold and \(\sigma\) a geodesic in \((M,g)\). Further, let \(P_\sigma (r)\) be a tube about \(\sigma\) with sufficiently small radius \(r\). The main result in this paper provide characterizations of complex space forms and locally Hermitian symmetric spaces by means of some extrinsic and intrinsic properties of the turbular hypersurfaces \(P_\sigma (r)\) relating to the shape operator in the Ricci operator on \(P_\sigma (r)\). These results are derived by using Jacobi vector fields and power series expansions for the above mentioned operators.