In this chapter, we begin to consider the invariant that truly characterizes differential geometry – curvature. The original formal definition of the curvature tensor, in §I.4, gives little hint to its profound geometric meaning; nevertheless, we indicated there the direction in which we are most interested in studying curvature – Jacobi's equation. In the Riemannian case, the torsion tensor of the Levi-Civita connection vanishes identically, so the curvature is the exclusive influence in studying the behavior of geodesics neighboring a given geodesic.

This, of course, is not the historical origin of curvature. In the beginning of differential geometry (i.e., in the beginning of the nineteenth century), it was viewed from the perspective of immediate human experience. Namely, the curvature of a curve attempted to measure the deviation of a curve in a plane or in space from being a straight line, and the various studies of curvature of a surface situated in space attempted to express how the surface deviated from being a plane in space. C. F. Gauss (1825, 1827) was the first to realize that one aspect of curvature, what we refer to as the Gauss curvature, did not depend on how the surface is situated in Euclidean space; that if the surface was bent – that is, deformed in such a manner as to preserve the measurement of lengths and angles in the surface – then while some curvatures were changed, other curvatures (namely, the Gauss curvature) were left invariant under the bending.