Curvature-homogeneous pseudo-Riemannian manifold - Revision history

Vipul: 2 revisions

2008-05-18T19:36:45Z

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Vipul at 18:34, 14 June 2007

2007-06-14T18:34:20Z

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	==Definition==		==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>		<math>\phi^k (\nabla^i R(p)) = \nabla^i R(q)</math>

Vipul at 18:33, 14 June 2007

2007-06-14T18:33:56Z

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{{pseudo-Riemannian metric property}}

==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:

<math>\phi^k (\nabla^i R(p)) = \nabla^i R(q)</math>

where <math>\nabla</math> denotes the covariant derivative with respect to the [[Levi-Civita connection]] and <math>R</math> 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)''