Geometry of the Riemann, Ricci, and Weyl Tensors

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Hi, I was wondering if someone wouldn't mind breaking down the geometrical differences between the Riemann, Ricci, and Weyl tensor. My current understanding is that the Ricci tensor describes the change in volume of a n-dimensional object in curved space from flat Euclidean space and that if we have a vanishing Weyl tensor that the space is conformally flat. However, I have a feeling these are 'rough' understandings and would just like to have a more solid concept of them. Oh, and in addition to these 3 tensors, does the Ricci scalar describe something else altogether? Thanks for any clarification!
 

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robphy
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I am also interested in this question
...but haven't found satisfactory answers.

Here are some references which may be helpful:
http://arxiv.org/abs/gr-qc/0103044 ("The Meaning of Einstein's Equation" by John Baez)
http://www.springerlink.com/content/j534310782m58575/ ("Geometry in a manifold with projective structure" by J. Ehlers and A. Schild)
http://www.springerlink.com/content/g1v07h0353723765/ ("The geometry of free fall and light propagation" by Jürgen Ehlers, Felix A. E. Pirani and Alfred Schild)
http://www.springerlink.com/content/q334654473650828/ ("On the physical significance of the Riemann tensor" Felix Pirani)

One specific question is "How would a pure mathematician interpret these tensors in the Riemannian setting [in arbitrary dimensions]?" and how would they compare and contrast with the Lorentzian-signature interpretations.
 

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