Characterizations of complex space forms by means of geodesic spheres and tubes
Volume 71 / 1996
Colloquium Mathematicum 71 (1996), 253-262
DOI: 10.4064/cm-71-2-253-262
Abstract
We prove that a connected complex space form ($M^n$,g,J) with n ≥ 4 can be characterized by the Ricci-semi-symmetry condition $\tilde{R}_{XY}·\tilde{ϱ}=0$ and by the semi-parallel condition $\tilde{R}_{XY}·σ=0$, considering special choices of tangent vectors $X,Y$ to small geodesic spheres or geodesic tubes (that is, tubes about geodesics), where $\tilde{R}$, $\tilde{ϱ}$ and $σ$ 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 $\tilde{R}_{XY}$ acts as a derivation.
