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Differential Geometry of Curves and Surfaces

Thomas F. Banchoff and Stephen T. Lovett
Chapman & Hall/CRC
Publication Date: 
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Fernando Q. Gouvêa
, on

See our review of the first edition. The authors say in their preface that they “preserved the intent and attempted to improve on the execution.” Apart from improvements on the exposition and new exercises and projects, the second edition has been reorganized to allow instructors to get to the Gauss-Bonnet Theorem more quickly. New sections discuss applications to cartography, hyperbolic and spherical geometry as examples of intrinsic geometry, and a discussion of curves and surfaces in n-dimensional Euclidean space.

Plane Curves: Local Properties
Position, Velocity, and Acceleration
Osculating Circles, Evolutes, and Involutes
Natural Equations


Plane Curves: Global Properties
Basic Properties
Rotation Index
Isoperimetric Inequality
Curvature, Convexity, and the Four-Vertex Theorem


Curves in Space: Local Properties
Definitions, Examples, and Differentiation
Curvature, Torsion, and the Frenet Frame
Osculating Plane and Osculating Sphere
Natural Equations


Curves in Space: Global Properties
Basic Properties
Indicatrices and Total Curvature
Knots and Links


Regular Surfaces
Parametrized Surfaces
Tangent Planes and Regular Surfaces
Change of Coordinates
The Tangent Space and the Normal Vector
Orientable Surfaces


The First and Second Fundamental Forms
The First Fundamental Form
Map Projections (Optional)
The Gauss Map
The Second Fundamental Form
Normal and Principal Curvatures
Gaussian and Mean Curvature
Developable Surfaces and Minimal Surfaces


The Fundamental Equations of Surfaces
Gauss’s Equations and the Christoffel Symbols
Codazzi Equations and the Theorema Egregium
The Fundamental Theorem of Surface Theory


The Gauss–Bonnet Theorem and Geometry of Geodesics
Curvatures and Torsion
Gauss–Bonnet Theorem, Local Form
Gauss–Bonnet Theorem, Global Form
Geodesic Coordinates
Applications to Plane, Spherical and Elliptic Geometry
Hyperbolic Geometry


Curves and Surfaces in n-Dimensional Euclidean Space
Curves in n-Dimensional Euclidean Space
Surfaces in Rn


Appendix: Tensor Notation