One cannot start discussing Riemannian geometry without mention of the classics. By “the classics,” we refer to the essays of C. F. Gauss (1825, 1827) and B. Riemann (1854), to G. Darboux's summary treatise (1894) of the work of the nineteenth century (and beginning of the twentieth), and to E. Cartan's lectures (1946) in which the method of moving frames became a powerful exciting tool of differential geometry.

Nor may one forget to recommend to the reader the delightful discussion of differential geometry in D. Hilbert–S. Cohn-Vossen (1952).

H. Hopf's notes (1946, 1956) remain eminently readable. A very helpful collection of more current introductory essays is the MAA Studies volume edited by S. S. Chern (1989).

In addition, one should refer to the “introductory” five-volume opus of M. Spivak (1970) – wherein the practice of differential geometry is presented in loving detail.

Most recently, one has a definitive overview of the subject at the end of the twentieth century by M. Berger (2003).

Our treatment here is mostly inspired by, and follows in many respects, J. Milnor's elegant and exceptionally clear lecture notes Milnor (1963).

A short summary of the progression of ideas of this chapter is as follows.

Whereas one has, given a differentiable manifold, a natural differentiation of functions on the manifold, one does not have a naturally determined method of differentiation of vector fields on the manifold. Therefore, one considers all possibilities of such differentiation – connections on the manifold. Once one actually picks such a differentiation procedure (i.e., a connection), one determines differentiation of vector fields along paths in the manifold.