The generalized axiom of spheres

Some generalization is proposed for the axiom of spheres. A collection of Riemannian spaces is constructed which satisfy the generalized axiom of spheres, but do not satisfy the earlier-known axioms of submanifolds. The structure is found of the curvature tensor of manifolds satisfying the generalized axiom of spheres. This structure mostly resembles the structure of the curvature tensor of manifolds with the generalized axiom of planes.