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Geodesic spheres in a nonflat complex space form and their integral curves of characteristic vector fields

For Hopf hypersurfaces in a nonflat complex space form M n (c;C), integral curves of their characteristic vector fields are "nice" curves in the sense that their extrinsic shapes in M n (c; C) are Kahler circles. In this paper we mainly study geodesic spheres in a nonflat complex space form M n (c;C). On these geodesic spheres we classify smooth curves whose extrinsic shapes are Kahler circles in M n (c,C), c ≠ 0. We also give a characterization of complex space forms among Kahler manifolds by extrinsic shapes of integral curves of characteristic vector fields on their geodesic spheres.