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Differential Geometry

Marcelo Epstein
Publication Date: 
Number of Pages: 
Mathematical Engineering
[Reviewed by
P. N. Ruane
, on

This book is based on a short course on ‘Differential Geometry and Continuum Mechanics’ given by Marcelo Epstein at the International Centre of Mathematical Sciences in Edinburgh in June 2013. The course provided a guided tour of differential geometry for researchers and graduate students in science and engineering — many of whom had a particular interest in continuum mechanics.

Since the attendees were largely familiar with elementary differential geometry, the ‘basic notions’ discussed during the course were way above the undergraduate level. Hence, familiar topics, such as the Frenet-Serret equations and the Gauss map have been bypassed, and the term ‘curvature’ is only used to describe the coefficients of an Ehresmann connection of a horizontal distribution of the general fibre bundle.

The material is developed within a purely topological framework that is clearly outlined in the first chapter. Within the first 25 pages there is a cogent summary of the relevant notions of point set topology, manifolds, groups, fibre bundles and groupoids (all with the prefix ‘topological’). Almost simultaneously, one reads of the application of such ideas to a wide variety of physical situations. For example, I learned about quite a bit about proto-Galilean space-time, configuration spaces of mechanical systems, symmetries in general (e.g. tiling of a bathroom floor) and microstructure of continuous media.

Because differentiable manifolds form a natural framework for the study of continuum mechanics, the longest chapter of the book is devoted to the major theme of ‘Differential Constructs’. Beginning with familiar ideas of differentiable manifolds, tangent bundles, vector fields and flows and cotangent bundles, it reveals the heavier machinery of Lie groups, Lie brackets, distributions and connections.

Given the vast mathematical scope of this slender tract, one might anticipate an indigestible presentation — but this would be very far from the truth. For one thing, there are many illuminating interludes in which more general concepts are illustrated by means of concrete examples (usually in \(\mathbb{R}^3\)). Other features include the author’s elegant expository style and the use of nicely produced diagrams at salient points in the text.

Finally, there is an intriguing range of physical applications in the last chapter, which reveal some ingenious uses of the preceding abstractions. For example, Lagrange’s postulate is formulated with reference to the geodesics of a Levi-Civita connection and Hamiltonian systems are considered in terms of symplectic manifolds. There are also applications to Cauchy flux theory of continuum mechanics and the dislocation of an atomic lattice.

Overall, this book is a gold mine of aesthetically pleasing mathematical ideas, the presentation of which is highly inspirational.

This book reminded Peter Ruane of much of the work he did on differential geometry as a postgraduate student in the 1970s.

Topological constructs
Physical illustrations
Differential constructs
Physical illustrations