# Curvatures in differential geometry-interpretation

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.