Generalized normal homogeneous Riemannian metrics on spheres and projective spaces

In this paper, we develop new methods to study generalized normal homogeneous Riemannian manifolds. In particular, we obtain a complete classification of generalized normal homogeneous Riemannian metrics on spheres S-n. We prove that for any connected (almost effective) transitive on S-n compact Lie group G, the family of G-invariant Riemannian metrics on S-n contains generalized normal homogeneous but not normal homogeneous metrics if and only if this family depends on more than one parameters and n >= 5. Any such family (that exists only for n = 2k + 1) contains a metric gcan of constant sectional curvature 1 on S-n. We alsoprove that (S2k+1, gcan) is Clifford-Wolf homogeneous, and therefore generalized normal homogeneous, with respect to G (except the groups G = SU (k + 1) with odd k + 1). The space of unit Killing vector fields on (S2k+1, gcan) from Lie algebra g of Lie group G is described as some symmetric space (except the case G = U(k + 1) when one obtains the union of all complex Grassmannians in Ck+1).