Characterization of complex space forms in terms of geodesics and circles on their geodesic spheres (Q1428970)

Let \(M\) be a Kähler manifold of complex dimension \(n\) \((\geq 2)\) and let \(G_x(r)\) denote a geodesic sphere of radius \(r\) centered at \(x\in M\). In this study, the authors prove the following theorems: Theorem 1. For a Kähler manifold \(M\) of complex dimension \(n\) \((\geq 2)\), the following two conditions are equivalent. i) \(M\) is a complex space form. ii) At an arbitrary point \(x\in M\), for each geodesic sphere \(G_x(r)\) of sufficiently small radius \(r\), every geodesic on \(G_x(r)\) has constant structure torsion. Theorem 2. For a complex \(n\) \((\geq 2)\)-dimensional Kähler manifold \(M\) the following two conditions are equivalent. i) \(M\) is locally congruent to a complex Euclidean space. ii) At an arbitrary point \(x\in M\), for each geodesic sphere \(G_x(r)\) of sufficiently small radius \(r\), there exists \(\kappa =\kappa _{x,r}>0\) such that every circle of curvature \(\kappa \) on \(G_x(r)\) has constant first curvature as a curve in the ambient space \(M.\)