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Curves and Surfaces

Marco Abate and Francesca Tovena
Publication Date: 
Number of Pages: 
[Reviewed by
P. N. Ruane
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This is an impeccably translated version of a book that was first published in Italy in 2006. In a single volume it provides several possible routes into differential geometry. For example, it could be used as a follow-up to a less rigorous introductory course on differential geometry. Indeed, almost one quarter of its 400 pages are devoted to ‘supplementary material’ of an advanced nature.

The minimal path (optimistically said by the authors to be appropriate for 2nd year students and onwards) consists of:

  • Chapter 1: local theory of curves in the plane and in \(\mathbb{R}^3\), focusing upon the central concepts of curvature and torsion.
  • Chapter 3: an introduction to the local theory of surfaces in \(\mathbb{R}^3\) (smooth functions, tangent planes, vectors and derivations, etc.)
  • Chapter 4: a deeper excursion into surface curvature involving the first and second fundamental forms, area, orientability and normal curvature.

This introduction to differential geometry terminates with an examination of Gauss’ Theorema Egregium, which (in the authors’ experience) nicely rounds off a two-month course. For a one-semester course, a selection from the following additional material is recommended:

  • Chapter 2: the global theory of curves (tubular neighbourhoods, Jordan Curve Theorem, Turning Tangents theorem, etc.)
  • Chapter 5: vector fields on a surface (integral curves, parallel transport)
  • Chapter 6: Gauss-Bonnet OR…
  • Chapter 7: classification of closed surfaces with constant Gaussian curvature.

To my mind, the level at which the material is presented means that readers who haven’t taken a course on advanced calculus or real analysis will struggle. Moreover, they will benefit greatly from having done a course on introductory topology (point set and algebraic). Lastly, although not specified by the authors, some knowledge of linear algebra will be of help.

What I like about this book is the provision of analytic and topological underpinnings of geometrical ideas. And, although this may compound the complexity of the narrative, it provides very strong foundations for readers who subsequently wish to continue beyond this book. Indeed, the nature of the supplementary material is such that readers will have a good idea as to the broader foundations upon which differential geometry rests. For example;

  • Chapter 1: Whitney’s Theorem, classification of submanifolds
  • Chapter 2: Four Vertex Theorem, Schönflies Theorem.
  • Chapter 5: Hopf-Rinow theorem.
  • Chapter 6: Existence of triangulations.

Another of the book’s major strengths is the clarity of its explanation and its mathematical accuracy. Then there are over 300 graded exercises preceded by many carefully worked examples.

In short, I can’t speak too highly of this book; and it certainly challenges my view that the best route into differential geometry is via differential forms. However, although I’ve no idea as to the standard of 2nd year students in Italian universities, I think it might be too much of a challenge for anyone other than senior undergraduate maths majors and above.

Peter Ruane was introduced to differential geometry over forty years ago, by means of Barrett O’Neill’s classic text. That approach was based upon (you’ve guessed it!) differential forms.