A Comprehensive Introduction to Differential Geometry, Volume 2

by Michael Spivak

Michael Spivak, Spivak, Michael

Genres: Mathematics, Geometry

375 pages, Hardcover

First published August 15, 2005

About the author

Michael David Spivak is a mathematician specializing in differential geometry, an expositor of mathematics, and the founder of Publish-or-Perish Press. He is the author of the five-volume Comprehensive Introduction to Differential Geometry. He received a Ph.D. from Princeton University under the supervision of John Milnor in 1964.

His book Calculus takes a very rigorous and theoretical approach to introductory calculus. It is used in calculus courses, particularly those with a pure mathematics emphasis, at many universities.

Spivak's book Calculus on Manifolds (often referred to as little Spivak) is also rather infamous as being one of the most difficult undergraduate mathematics textbooks.

Reviews

[Woflmao · Rating 5 out of 5]

This is the book you wish you had first learned differential geometry from. It is all about curvature. The first few chapters are mostly classical results on the curvature of curves in space. It then takes a historical approach to developing the concept of Gauß curvature of a surface/manifold. Beginning with the classical article by Gauß himself, "Disquisitiones Generales circa Superficies Curvas", in English, of course, and with a helpful commentary by Spivak on the side. Curvature is not something that can be easily visualised by handwaving and a few intuitive explanations. This is followed by Riemann's short article in which he proposes the generalisation of Gauß' work to what is now called manifolds, and develops the curvature tensor. Even in this careful development by Spivak along the historical lines, it is clear that most of the main properties of curvature are only obtained by difficult calculations. In light of this, Gauß' and Riemann's ideas seem all the more brilliant, as they had hardly any inutition to cling to when developing them.
Once the curvature on Riemannian manifolds has been studied, Spivak takes a tour through different alternative approaches to defining connections and curvature on manifolds in the second half of the book. This includes the connections given by defining a covariant derivative operator à la Koszul, the method of moving frames by Cartan and finally the most general definition of connections on principal bundles due to Ehresmann. Spivak takes great care in comparing the different definitions and showing where they lead to equivalent results, and where they one is more general than the other (especially since not all connections arise from Riemannian metrics). This second half of the book is not an easy read, due to the aforementioned technical character of the theory. But I dare say that one can hardly find more motivated and careful explanations of the subject.