Differential geometry has a wide range of applications, going far beyond strictly mathematical pursuits to include architecture, engineering, and just about every scientific discipline. John Oprea’s second edition of Differential Geometry and Its Applications illuminates a wide range of ideas that can be beneficial to students majoring not only in mathematics but also in other fields.
The textbook touches on many different mathematical concepts, including aspects of linear algebra, the Gauss-Bonnet Theorem, and geodesics. It also encourages students to visualize and experiment with the ideas they are studying through their use of the computer program Maple. This allows students to develop a better understanding of the mathematics involved in differential geometry.
Note to Students
1. The Geometry of Curves
4. Constant Mean Curvature Surfaces
5. Geodesics, Metrics and Isometries
6. Holonomy and the Gauss-Bonnet Theorem
7. The Calculus of Variations and Geometry
8. A Glimpse at Higher Dimensions
List of Examples
Hints and Solutions to Selected Problems
Suggested Projects for Differential Geometry
About the Author
John Oprea was born in Cleveland, Ohio and was educated at Case Western Reserve University and at Ohio State University. He received his PhD at OSU in 1982 and, after a post-doc at Purdue University, he began his tenure at Cleveland State in 1985. Oprea is a member of the Mathematical Association of America and the American Mathematical Society. He is an Associate Editor of the Journal of Geometry and Symmetry in Physics. In 1996, Oprea was awarded the MAA’s Lester R. ford award for his Monthly article, “Geometry and the Foucault Pendulum.” Besides various journal articles on topology and geometry, he is also the author of The Mathematics of Soap Films (AMS Student Math Library, volume 10), Symplectic Manifolds with no Kähler Structure (with A. Tralle, Springer Lecture Notes in Mathematics, volume 166). Lusternik-Schnirelmann Category (with O. Cornea, G. Lupton and D. Tanré, AMS Mathematical Surveys and Monographs, volume 103) and the forthcoming Algebraic Models in Geometry (with Y. Felix and D. Tanré, for Oxford University Press).
John Oprea begins Differential Geometry and Its Applications with the notion that differential geometry is the natural next course in the undergraduate mathematics sequence after linear algebra. He argues that once students have studied some multivariable calculus and linear algebra, a differential geometry course provides an attractive transition to more advanced abstract or applied courses. His thoughtful presentation in this book makes an excellent case for this. As he says, the natural progression of concepts in differential geometry allows the student to progress gradually from calculator to thinker.
This edition of the text is over a hundred pages longer than the first edition. Evidently Oprea has incorporated many suggestions from those who have taught from the text. There is a good deal to like about this book: the writing is lucid, drawings and diagrams are plentiful and carefully done, and the author conveys a contagious sense of enthusiasm for his subject. Continued...