Geometry of Manifolds

Rating:
Summary: Spectacular geometric insight into differential geometry
Review: As a differential geometer for the past 30 years, I own 8 introductions to the field, and I have perused a half-dozen others. Bishop & Crittenden's "Geometry of Manifolds" is unparallelled for imparting a strong geometric intuition about Lie derivatives and connections on fiber bundles, which is the key to understanding this field, plus general relativity, Einstein-Cartan theory, and gauge theories in physics. Bishop & Crittenden taught me to see pictures for the main constructions and theorems, though I admit I had to work hard to build my intuition. It is particularly strong in establishing a 1-1 correspondence between geometric images of horizontal planes in fiber bundles and the language of differential forms in fiber bundles. I first read the classic Chevalley's "Lie Groups" which is very strong on geometric intuition about manifolds and Lie groups. I also relied on Kobayashi & Numizu's "Foundations of Differential Geometry" I & II because ity has better algebraic and computational treatment in some areas, and it cover topics not in "Geometry of Manifolds."

If you are really serious about differential geometry, if you prefer to think in terms of geometric visualization when possible and use algebra afterwards, then this book is the greatest -- nothing I have seen comes close. If you prefer to think mainly with abstract algebra, then you might prefer Kobayashi and Nomizu.

Richard J. Petti