Characterizations of complex space forms by means of geodesic spheres and tubes

We prove that a connected complex space form (${\displaystyle M^n}$,g,J) with n ≥ 4 can be characterized by the Ricci-semi-symmetry condition ${\displaystyle {\tilde{R}}_{XY}·\tilde{\varrho }=0}$ and by the semi-parallel condition ${\displaystyle {\tilde{R}}_{XY}·\sigma =0}$, considering special choices of tangent vectors ${\displaystyle X,Y}$ to small geodesic spheres or geodesic tubes (that is, tubes about geodesics), where ${\displaystyle \tilde{R}}$, ${\displaystyle \tilde{\varrho }}$ and ${\displaystyle \sigma }$ denote the Riemann curvature tensor, the corresponding Ricci tensor of type (0,2) and the second fundamental form of the spheres or tubes and where ${\displaystyle {\tilde{R}}_{XY}}$ acts as a derivation.