GEODESIC SPHERES AND 2-POINT HOMOGENEOUS SPACES

In the Osserman conjecture and in the isoparametric conjecture it is s tated that two-point homogeneous spaces may be characterized via the c onstancy of the eigenvalues of the Jacobi operator or the shape operat or of geodesic spheres, respectively. These conjectures remain open, b ut in this paper we give complete positive results for similar stateme nts about other symmetric endomorphism fields on small geodesic sphere s. In addition, we derive more characteristic properties for this clas s of spaces by using other properties of small geodesic spheres. In pa rticular, we study Riemannian manifolds with (curvature) homogeneous g eodesic spheres.