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Differential geometry of curves and surfaces. (English) Zbl 1200.53001

Natick, MA: A K Peters (ISBN 978-1-56881-456-8/hbk). xvi, 331 p. (2010).
This textbooks covers the content of a one-semester undergraduate course in classical differential geometry, starting with planar curves and ending with the global version of the Gauss-Bonnet Theorem. The table of contents is given below.
It seems to be the author’s understanding that an intuitive and visual introduction to the subject is beneficial in an undergraduate course. (The reviewer shares this point of view.) This attitude is reflected in the text. The authors spent quite some time on motivating particular concepts and discuss simple but instructive examples. At the same time, they do not neglect rigour and precision.
This book is accompanied by [St. T. Lovett, Differential Geometry of Manifolds. Natick, MA: A K Peters (2010; Zbl 1205.53001)], a textbook that represents the modern view on differential geometry. According to the authors, neither book is a pre-requisite for the other but reading one will be helpful for reading the other. In fact, especially in the later parts of the book under review the gentle guidance towards differential geometry on manifolds is clearly visible. The second book in the pair is also often references when more advanced results from calculus or topology are needed.
Each section concludes with a collection of exercises (but no solutions). Approximately 100 figures, a list of 31 bibliography items and an index complete the printed part of the book. As a distinguishing feature to other textbooks, there is an accompanying web-page http://www.akpeters.com/DiffGeo/ containing numerous interactive Java applets. Whenever a definition or an example is illustrated or visualized by an applet, this is indicated by a special symbol in the outer margin. The applets are well-suited for use in classroom teaching or as an aid to self-study.
We conclude this review with the book’s table of contents:
Preface
Acknowledgements
1. Plane Curves: Local Properties
Parameterizations
Position, Velocity, and Acceleration
Curvature
Osculating Circles, Evolutes, and Involutes
Natural Equations
2. Plane Curves: Global Properties
Basic Properties
Rotation Index
Isoperimetric Inequality
Curvature, Convexity, and the Four-Vertex Theorem
3. Curves in Space: Local Properties
Definitions, Examples, and Differentiation
Curvature, Torsion, and the Frenet Frame
Osculating Plane and Osculating Sphere
Natural Equations
4. Curves in Space: Global Properties
Basic Properties
Indicatrices and Total Curvature
Knots and Links
5. Regular Surfaces
Parametrized Surfaces
Tangent Planes and Regular Surfaces
Change of Coordinates
The Tangent Space and the Normal Vector
Orientable Surfaces
6. The First and Second Fundamental Forms
The First Fundamental Form
The Gauss Map
The Second Fundamental Form
Normal and Principal Curvatures
Gaussian and Mean Curvature
Ruled Surfaces and Minimal Surfaces
7. The Fundamental Equations of Surfaces
Tensor Notation
Gauss’s Equations and the Christoffel Symbols
Codazzi Equations and the Theorema Egregium
The Fundamental Theorem of Surface Theory
8. Curves on Surfaces
Curvatures and Torsion
Geodesics
Geodesic Coordinates
Gauss-Bonnet Theorem and Applications
Intrinsic Geometry
Bibliography

MSC:

53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
53A04 Curves in Euclidean and related spaces
53A05 Surfaces in Euclidean and related spaces

Citations:

Zbl 1205.53001
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