Differential Geometry of Curves and Surfaces

Preface; Acknowledgements; Plane Curves: Local Properties; Parameterizations; 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; The Gauss Map; The Second Fundamental Form; Normal and Principal Curvatures; Gaussian and Mean Curvature; Ruled Surfaces and Minimal Surfaces; 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; Curves on Surfaces; Curvatures and Torsion; Geodesics; Geodesic Coordinates; Gauss-Bonnet Theorem and Applications; Intrinsic Geometry; Bibliography

Thomas F. Banchoff is a geometer and has been a professor at Brown University since 1967. Banchoff was president of the MAA from 1999-2000. He is published widely and known to a broad audience as editor and commentator on Abbott's Flatland. He has been the recipient of such awards as the MAA National Award for Distinguished College or University Teaching of Mathematics and most recently the 2007 Teaching with Technology Award. Stephen Lovett is an associate professor of mathematics at Wheaton College in Illinois. Lovett has also taught at Eastern Nazarene College and has taught introductory courses on differential geometry for many years. Lovett has travelled extensively and has given many talks over the past several years on differential and algebraic geometry, as well as cryptography.