You are here

Differential Geometry of Curves and Surfaces

Thomas F. Banchoff and Stephen T. Lovett
Publisher: 
Chapman & Hall/CRC
Publication Date: 
2015
Number of Pages: 
414
Edition: 
2
Price: 
69.95
ISBN: 
9781482247343
Category: 
Textbook
BLL Rating: 

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Fernando Q. Gouvêa
, on
09/28/2015
]

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
Parametrizations
Position, Velocity, and Acceleration
Curvature
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
Geodesics
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