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)