Talk:Curvature of Riemannian manifolds

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Sectional curvature part, first paragraph: there are some missing symbols. Charles Matthews 07:58, 13 May 2004 (UTC)


ok?
Tosha 20:46, 13 May 2004 (UTC)

If you remove the other article OK, but still I doubt that you catch the most general aspect of curvature in the sense of Nomizu, Kulkarni and other modern writers. Look at the reference http://www.EarningCharts.NET/ipm/ipmWaves.htm where you find more references. In the references there (look also at that one in Lecture Notes in Mathematics) you find a decomposition of the space of all curvature structures in terms of Lie and Jordan algebras. And you find how elegantly electrodynamics and gravitational waves fit into the curvature play, look at the basic work of Lichnerowics and the reference given below. As an ,algebraiker' I like to write the curvature structure in the following triple form, generalizing the concept of Lie triples (the book of Otmar Loos giving a nice generalization of Lie theory): [x,y,z]=R(x,y)z. This concept generalizes the notion of a Lie triple to that one of a curvature triple, where only the Jacobiidentity is missing, but a reference to the bilinear form <,> is added in such a way, that R(x,y) is an element of the pseudoorthogonal Lie algebra. Note that the complete work of Ricci, Einstein and Weyl can be summarized as a Levi-type decomposition of the space of curvature structures (the Levi decomposition for Lie algebras is the decomposition of a Lie algebra into a short exact sequence of Lie algebras, where the first non-trivial Lie algebra is a maximal solvable ideal and the third Lie algebra is the semisimple Levi factor - the second being the given Lie algebra itself). In the articles cited below it is shown, how to decompose the curvature (vector) space into the map of a semisimple Jordan algebra and a subspace, which may turn out to be a model of a ,solvable' ideal of the curvature structure. Actually this is the condensed content of the work of Ricci, Einstein and Weyl. All this shows, that we do not yet understand this curvature space completely. Especially the gravitational wave aspect needs clarification. This implies contributions to the running search for gravitational waves. Note that it is easy to see, that there are no gravitational waves if the bilinear form <,> is positive definite. Let me add the reference for this [Lecture Notes in Mathematics 1156 1978 p.316-337], the yellow Springer books, and [J.Math.Phys. 19 1978 p.1118-1125]. Hannes Tilgner

Please stay basisfree. There is no need to introduce a basis in order to define sectional and Ricci curvature.

It would be nice to add one more curvature formula since it shows more clearly how Ruv measures the lack of commutativity of second covariant derivatives in general:

R_{uv}=\nabla^2_{uv}-\nabla^2_{vu}

where \nabla^2_{uv} is the "second covariant derivative" tensor (easy to verify it's tensorial in u, v, and w):

\nabla^2_{uv}w\equiv\nabla_u\nabla_vw-\nabla_{\nabla_uv}w

The connection must be torsion-free for it to work which is fine as the article focuses on the Levi-Cività connection.

Another change worth introducing would be to cover the general case of indefinite metrics — this would make the article usable for those interested in general relativity calculations. Not many changes are needed, e.g. Cartan's curvature forms \Omega^i_{\ j} aren't antisymmetric in general (although \displaystyle\Omega_{ij} are). Instead \Omega^i_{\ j}=-\eta^{il}\eta_{jk}\Omega^k_{\ l} and similarly for the connection 1-forms, where η is the matrix of metric coefficients with respect to the moving frame (which is typically either orthonormal or null). JanBielawski 04:52, 11 August 2006 (UTC)

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