On a generalization of curvature homogeneous spaces
In 2011 the authors introduce the notion of curvature homogeneous spaces of type \((1,3)\). Let \(R\) be the \((1,3)\)-type curvature tensor and \(\mathcal R\) the corresponding curvature tensor of type \((0,4)\). The Riemannian manifold \(M\) is type \((1,3)\) curvature homogeneous if for any points \(p,q \in M\) there is a linear homothety \(f : T_p(M) \to T_q(M)\) such that \(f^\star (R_q) = R_p\). The authors consider this notion as a proper generalization of the curvature homogeneous manifolds (in which the curvature tensor of type \((0,4)\) is used) in the sense of general Spencer's considerations. In this paper the authors, following Spencer's theory, introduce the notion of higher order homogeneous Riemannian manifolds of type \((1,3)\). Instead of Spencer's conditions \(P(k)\) : For any points \(p,q \in M\) there is a linear isometry \(F:T_p(M) \to T_q(M)\) such that \(F^\star ((D^k \mathcal R)_q) = (D^k\mathcal R)_p\). The authors impose the following conditions: \(Q(k)\) : For any points \(p,q \in M\) there is a linear homothety \(h:T_p(M) \to T_q(M)\) such that \(h^\star (\mathcal (D^k R)_q) = (D^k R)_p\). A Riemannian manifold \((M,g)\) is said to be the curvature homogeneous up to order \(r\) and of type \((1,3)\) if the conditions \(Q(0), Q(1), \dots Q(r)\) are satisfied. In this paper some condition which is equivalent to the condition \(Q(k)\) is given. Then they give an example of a 3-dimensional Riemannian manifold which is type \((1,3)\) curvature homogeneous up to order \(1\) but not locally homogeneous. This result contrasts with the situation for the condition \(P(1)\) -- up to now there exist no examples of Riemannian manifolds satisfying \(P(1)\) which are not locally homogeneous.
scientific article; zbMATH DE number 6135237