Curvature-homogeneous pseudo-Riemannian manifold: Difference between revisions

Latest revision as of 19:36, 18 May 2008

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)

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 ==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>