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Differential Geometry of Curves and Surfaces

Thomas Banchoff and Stephen Lovett
A K Peters
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
William Porter
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Differential Geometry of Curves and Surfaces, by Thomas Banchoff and Stephen Lovett, presents a thorough introduction to the study of curves and surfaces. Geared toward advanced undergraduates, specifically those that are particularly calculus-savvy, this text carefully develops the rudiments of differential geometry by first examining plane curves, then space curves, moving on to regular surfaces, the first and second fundamental forms, and finally curves on regular surfaces.

One very satisfying characteristic of this book is the geometric intuition that the authors nurture through their careful exposition and examples. For instance, rather than simply defining space curves analogously to plane curves and moving on to torsion, Banchoff and Lovett present a long series of examples ranging from simple lines in three-space through the twisted cubic. Later, after curvature and torsion have been defined, they return to flesh out a couple of these examples. This type of exposition pervades most chapters of the book. It becomes particularly helpful when less intuitive concepts (e.g. the Christoffel symbols) are introduced.

I think the exercises in this book are wonderfully balanced. They range from the almost trivial computations to a carefully outlined exercise deriving the transformation rules for Christoffel symbols. The sections devoted to explaining the tensor notations more prevalent in physics textbooks are even equipped with problems that deal with physics: the moment of inertia tensor and the Minkowski metric.

It may be inevitable in this subject that the writing at times becomes rather dense. I found myself rereading sentences three or four times in order to grasp a definition or decide exactly what I was being told.

That being said, precision is key, and the authors split hairs in such ways as to emphasize very subtle and important differences where they exist. One example of this comes when they discuss diffeomorphisms. They spend a page discussing why one of the points in the definition is absolutely necessary, presenting examples to illustrate the idea.

Finally, though this book is written for “undergraduates,” it presupposes a huge amount of mathematical experience. Though it begins with elementary vector algebra, the difficulty jumps sharply somewhere around the middle of the book and rises steadily thereafter. Though I was still enjoying it greatly, I wasn’t sure it was the same book I’d begun reading. Beginning with such simple concepts as parametrizing lines in two-space is perhaps necessary, but it is deceptive as an index of how difficult the book becomes. I’d recommend a more than healthy understanding of linear algebra, as much calculus as possible, some ordinary differential equations, and a geometric understanding of point-set topology in Euclidean two- and three-space.

This book is an excellent, detailed introduction to differential geometry. It would, without a doubt, stand alone as the main text for its namesake undergraduate class. Further, I’d recommend this book to any mathematically minded physicist who is seeking an intuitive understanding of the methods characteristic of classical and relativistic mechanics.

William Porter is an undergraduate mathematics-physics double major at a small liberal arts college. He enjoys ballroom dancing, cooking, and T. S. Eliot’s poetry. He thinks that Rachmaninoff writes amazing music and rain is the best weather. His favorite Big Bang Theory character is Sheldon.

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