Extrinsic shape of circles and standard imbeddings of projective spaces (Q1360943)

The authors obtain new geometric characterization of the first standard imbedding of projective spaces. For instance, they prove the following: Let \(M\) be a non-flat Kähler manifold of real dimension \(2n\) \((\geq 4)\) which is immersed into a real space form \(\widetilde M^{2n+p}(c)\). If there exists \(k>0\) such that all holomorphic circles of curvature \(k\) are circles in \(\widetilde M^{2n+p}(c)\), then \(M\) is locally congruent to a complex projective space imbedded into some sphere in \(\widetilde M^{2n+p}(c)\) through the first standard imbedding. A similar result was also obtained for quaternionic projective spaces.