Geodesic spheres and two-point homogeneous spaces

In the Osserman conjecture and in the isoparametric conjecture it is stated that two-point homogeneous spaces may be characterized via the constancy of the eigenvalues of the Jacobi operator or the shape operator of geodesic spheres, respectively. These conjectures remain open, but in this paper we give complete positive results for similar statements about other symmetric endomorphism fields on small geodesic spheres. In addition, we derive more characteristic properties for this class of spaces by using other properties of small geodesic spheres. In particular, we study Riemannian manifolds with (curvature) homogeneous geodesic spheres.