3.1 A parallel vector field in R2 is just a constant field. Now, on a surface, there are generally no (even local) parallel vector fields. How much the parallel transport of a field along a small closed curve differ from the identity is measured in terms of the curvature of the surface, a function k: M → R. Now, on an n-dimensional manifold, the effect of the parallel transport along small closed curves lying in different “2-planes” depends on these very planes, and actually involves a (1, 3)-tensor, the curvature tensor of the Riemannian manifold.
KeywordsRiemannian Manifold Sectional Curvature Curvature Tensor Constant Curvature Ricci Curvature
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