Enhanced descriptions from Syndetics:

From the coauthor of Differential Geometry of Curves and Surfaces, this companion book presents the extension of differential geometry from curves and surfaces to manifolds in general. It provides a broad introduction to the field of differentiable and Riemannian manifolds, tying together the classical and modern formulations. The three appendices provide background information on point set topology, calculus of variations, and multilinear algebra--topics that may not have been covered in the prerequisite courses of multivariable calculus and linear algebra.

Differential Geometry of Manifolds takes a practical approach, containing extensive exercises and focusing on applications of differential geometry in physics, including the Hamiltonian formulation of dynamics (with a view toward symplectic manifolds), the tensorial formulation of electromagnetism, some string theory, and some fundamental concepts in general relativity.

Reviews provided by Syndetics

CHOICE Review

Banchoff (Brown Univ.) and Lovett (Wheaton College) have produced two connected works on modern differential geometry, assuming only minimal prerequisites. The first volume, Differential Geometry of Curves and Spaces, is a rich, concrete introduction to the theory of plane and space curves and surfaces in R3, full of classical topics that undergraduate students all too rarely see. Curves includes local notions such as curvature, torsion, and the Frenet frame, as well as global results such as the four-vertex theorem for plane curves and the Fary-Milnor theorem about knottedness of space curves. The study of surfaces requires development of more machinery and, accordingly, the authors begin with local definitions using coordinate patches, and then introduce the Gauss map, fundamental forms, geodesics, varieties of curvature, and treatments of ruled and minimal surfaces. There is a lucid discussion of tensors, phrased in local coordinate terms, leading to structural equations for surfaces and the Theorema Egregium. The volume culminates with the Gauss–Bonnet theorem and some of its consequences. What makes this book so special, however, are the dozens of illuminating online applets that accompany it. Readers can readily use these to visualize and interact with examples from the text, or to develop their own.Differential Geometry of Manifolds, by Lovett alone, continues the first title, although it can be read independently. Intended to provide a working understanding of the differential geometry of n-dimensional manifolds, it does a good deal more, offering treatments of analysis on manifolds (including the generalized Stokes's theorem) in addition to Riemannian geometry. An especially interesting chapter on applications to physics includes some general relativity, string theory, symplectic geometry, and Hamiltonian mechanics. Appendixes give background in general topology, the calculus of variations, and multilinear algebra. (One small awkwardness: the Banchoff and Lovett volume frequently refers to the appendix on topology in Lovett's work.)Both books are very carefully constructed and written with a deft touch and an enticing, friendly tone. The stated prerequisites of multivariable calculus and linear algebra are perhaps ambitious for undergraduates in places, but these works certainly would motivate readers to make the necessary investment of time and thought. These two books are valuable library acquisitions. Summing Up: Highly recommended. Upper-division undergraduates and graduate students. S. J. Colley Oberlin College

Author notes provided by Syndetics

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 traveled extensively and has given many talks over the past several years on differential and algebraic geometry, as well as cryptography.