Curvature-homogeneous pseudo-Riemannian manifold: Difference between revisions

Revision as of 18:34, 14 June 2007

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 $ϕ : T_{p} (M) \to T_{q} (M)$ such that for all integers $i \in {0, 1, 2, \dots, k}$ the following holds:

$ϕ^{k} (\nabla^{i} R (p)) = \nabla^{i} R (q)$

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)

@@ Line 3: / Line 3: @@
 ==Definition==
-A [[pseudo-Riemannian manifold]] <math>(M,g)</math> is said to be '''curvature-homogeneous''' up to order <math>k</math> if for any points <math>p, q \in M</math> there exists a linear isometry <math>\phi:T_p(M) \to \T_q(M)</math> such that for all integers <math>i \in \{ 0,1,2, \ldots, k \}</math> the following holds:
+A [[pseudo-Riemannian manifold]] <math>(M,g)</math> is said to be '''curvature-homogeneous''' up to order <math>k</math> if for any points <math>p, q \in M</math> there exists a linear isometry <math>\phi:T_p(M) \to T_q(M)</math> such that for all integers <math>i \in \{ 0,1,2, \ldots, k \}</math> the following holds:
 <math>\phi^k (\nabla^i R(p)) = \nabla^i R(q)</math>