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The are various notions of curvatures in differential geometry: soft such as full curvature tensor for a given connection (which is tensor of type $(1,3)$), Ricci curvature tensor (type $(0,2)$ tensor) and also hard (requiring the choice of the metric) such as sectional curvature and scalar curvature. Each notion of curvature measures some kind of defect of flatness: this is very nice explained on wikipedia. However I would like to see the proofs that all of these curvatures have such a geometric interpretation: to be more precise: 1. The sectional curvature measures how geodesic trangles differ from euclidean ones.
2. Scalar curvature express the ratio between the volumes of geodesic balls on manifold and balls in euclidean space.
3. Ricci tensor expresses the relation between metric volume form on manifold and standard measure on $\mathbb{R}^n$.
4. Finally the full curvature tensor has something to do with so called holonomy (and parallel transport).
I will be very grateful for giving me some references.

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For (1), see here

For (2), see here

For (3), see here.

For (4), see here.

All of those articles also contain references to more fundamental work.

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  • $\begingroup$ As I mentioned in my post, I read these wiki articles: each of them contains a lot more information about discussed concept of curvature and obviously there are some references but I would like to know where precise I can found the results about the geometric interpretation. For example for the scalar curvature, I haven't found the answer in Kobayashi and Nomizu's book, although it is given on wikipedia article as a reference source. $\endgroup$ – truebaran Aug 10 '15 at 16:04
  • $\begingroup$ Fair enough. To be honest, I read your question rather too quickly and, being a bit busy, went for the fastest answer I could possibly give. I don't remember other sources at the moment. Besides, being a physicist, my answer(s) and references might not be as satisfactory to you as other people's. $\endgroup$ – wltrup Aug 10 '15 at 16:07

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