Curvature-homogeneous pseudo-Riemannian manifold

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Template:Pseudo-Riemannian metric property

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)