An Introduction to Riemannian Geometry

Godinho and Natário are very explicit as regards what they’re about in this book: in their Preface they state that the book is meant for a “one semester course … [for] mathematics, physics, and engineering students … [and provides] a quick introduction to differential geometry, including differential forms, followed by the main ideas of Riemannian geometry (including minimizing properties of geodesics, completeness, and curvature).” The book’s last two chapters proper are more discursive, or perhaps eclectic, in that they are concerned with “possible applications,” the foci being geometric mechanics and general relativity. Godinho and Natário note that these chapters have in fact been used for autonomous one-semester courses, presumably succeeding the (solid) introduction to Riemannian geometry that is part and parcel of the book’s first four chapters.

It is noteworthy that the authors single out the three themes mentioned parenthetically above, i.e. minimizing properties of geodesics, completeness, and curvature, as characterizing the heart of Riemannian geometry at its “first-pass” level. Consider in this connection what Milnor does in his unsurpassed book on Morse Theory, in preparation for discussing Smale’s revolutionary geometric approach to the subject: his second chapter, titled “A rapid course in Riemannian geometry” (and rapid it is: 23 pages), features covariant differentiation, the curvature tensor, and geodesics and completeness. After that it’s on to “The calculus of variations applied to geodesics” and we’re soon hitting some deep stuff (and pretty avant garde in 1963, when Milnor wrote the book).

The point is, of course, that regardless of the context, whether it be Milnor’s approach to Morse theory or the physics applications that Godinho and Natário aim for, there is an objective, non-negotiable core of Riemannian geometry to be covered: in the book under review we find covariant differentiation covered in the section on affine connections (in Chapter 3), as it must be, of course, with geodesics and completion covered nearby; and curvature makes its appearance in Chapter 4.

A propos, the apotheosis of the material on geodesics and completeness is the Hopf-Rinov theorem (p. 118): Suppose we have a connected smooth manifold \(M\) equipped with a Riemannian inner product \(g\), and we define a distance on \(M\) via the rule that the shortest distance between two points is the infimum of the lengths of the piecewise differentiable curves connecting these points (of course here the distance formula involves \(g\) in a critical way). Then we say that the Riemannian manifold \((M,g)\) is geodesically complete if and only if we can exponentiate locally throughout the tangent bundle to the manifold, i.e. we get an exponential map from every tangent space \(T_pM\) to the manifold (i.e. at each \(p\in M\)). Then it is in fact the case that geodesic completeness is equivalent to Cauchy completeness for \(M\) relative to the aforementioned distance function. Obviously this result, by itself, is worth the price of admission.

The rest of the book is, briefly, as follows: Chapters 1 and 2 comprise a solid introduction to differentiable manifolds and differential forms, with Stokes’ theorem and the matter of orientation and the volume form closing Chapter 2. Chapter 5 is on geometric mechanics (holonomy, Lagrangian mechanics, and Hamiltonian mechanics appear here), and Chapter 6 is devoted to Einstein’s relativity, going all the way up to recent stuff on singularities by Hawking and by Penrose. And, by the way, Gauss-Bonnet is found in the middle of Chapter 4.

The book is also pedagogically very sound: all of the well-over 100 pages of Chapter 7 are devoted to solutions to selected exercises from the text proper. This book is a very nice and valuable addition to the literature which serves its purpose well: it’s a clear and quite accessible introduction to an always hugely important subject with, if you’ll pardon the pun, many connections.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.