# Curvature-homogeneous pseudo-Riemannian manifold

## Definition

A pseudo-Riemannian manifold $(M,g)$ is said to be curvature-homogeneous up to order $k$ if for any points $p, q \in M$ there exists a linear isometry $\phi:T_p(M) \to T_q(M)$ such that for all integers $i \in \{ 0,1,2, \ldots, k \}$ the following holds:

$\phi^k (\nabla^i R(p)) = \nabla^i R(q)$

where $\nabla$ denotes the covariant derivative with respect to the Levi-Civita connection and $R$ denotes the Riemann curvature tensor.

A pseudo-Riemannian manifold is simply termed curvature-homogeneous if it is curvature-homogeneous up to order zero.

## References

• Infinitesimally homogeneous spaces by I M Singerm Commun. Pure Appl. Math. 13 685 - 97 (1960)