Differential Geometry of Curves and Surfaces

Thomas F. Banchoff is a geometer and a professor at Brown University. Dr. Banchoff was president of the Mathematical Association of America (MAA) from 1999 to 2000. He has published numerous papers in a variety of journals and has been the recipient of many honors, including the MAA’s Deborah and Franklin Tepper Haimo Award and Brown’s Teaching with Technology Award. He is the author of several books, including Linear Algebra Through Geometry with John Wermer and Beyond the Third Dimension. Stephen Lovett is an associate professor of mathematics at Wheaton College. Dr. Lovett has taught introductory courses on differential geometry for many years, including at Eastern Nazarene College. He has given many talks over the past several years on differential and algebraic geometry as well as cryptography. In 2015, he was awarded Wheaton’s Senior Scholarship Faculty Award. He is the author of Abstract Algebra: Structures and Applications, Differential Geometry of Manifolds, Second Edition, A Transition to Advanced Mathematics with Danilo Dedrichs (forthcoming), all published by CRC Press.

Preface
1 Plane Curves: Local Properties
1.1 Parametrizations
1.2 Position, Velocity, and Acceleration
1.3 Curvature
1.4 Osculating Circles, Evolutes, Involutes
1.5 Natural Equations
2 Plane Curves: Global Properties
2.1 Basic Properties
2.2 Rotation Index
2.3 Isoperimetric Inequality
2.4 Curvature, Convexity, and the Four-Vertex Theorem
3 Curves in Space: Local Properties
3.1 Definitions, Examples, and Differentiation
3.2 Curvature, Torsion, and the Frenet Frame
3.3 Osculating Plane and Osculating Sphere
3.4 Natural Equations
4 Curves in Space: Global Properties
4.1 Basic Properties
4.2 Indicatrices and Total Curvature
4.3 Knots and Links
5 Regular Surfaces
5.1 Parametrized Surfaces
5.2 Tangent Planes; The Differential
5.3 Regular Surfaces
5.4 Change of Coordinates; Orientability
6 First and Second Fundamental Forms
6.1 The First Fundamental Form
6.2 Map Projections (Optional)
6.3 The Gauss Map
6.4 The Second Fundamental Form
6.5 Normal and Principal Curvatures
6.6 Gaussian and Mean Curvatures
6.7 Developable Surfaces; Minimal Surfaces
7 Fundamental Equations of Surfaces
7.1 Gauss’s Equations; Christoffel Symbols
7.2 Codazzi Equations; Theorema Egregium
7.3 Fundamental Theorem of Surface Theory
8 Gauss-Bonnet Theorem; Geodesics
8.1 Curvatures and Torsion
8.2 Gauss-Bonnet Theorem, Local Form
8.3 Gauss-Bonnet Theorem, Global Form
8.4 Geodesics
8.5 Geodesic Coordinates
8.6 Applications to Plane, Spherical, and Elliptic Geometry
8.7 Hyperbolic Geometry
9 Curves and Surfaces in n-dimensional Space
9.1 Curves in n-dimensional Euclidean Space
9.2 Surfaces in Euclidean n-Space
Appendix A: Tensor Notation
Index