  Osserman manifolds are a generalization of locally two-point homogeneous
spaces. We introduce $k$-root manifolds in which the reduced Jacobi operator
has exactly $k$ eigenvalues. We investigate one-root and two-root manifolds as
another generalization of locally two-point homogeneous spaces. We prove that
there is no two-root Riemannian manifold of odd dimension. In twice an odd
dimension, we describe all two-root Riemannian algebraic curvature tensors and
give additional conditions for two-root Riemannian manifolds.
