Foundations of Differential Geometry is an influential 2-volume mathematics book on differential geometry written by Shoshichi Kobayashi and Katsumi Nomizu. The first volume was published in 1963 and the second in 1969, by Interscience Publishers. Both were published again in 1996 as Wiley Classics Library.
These books have received multiple reviews throughout which three themes repeatedly occur.
It is the definitive book. Reviews say it is considered to be "THE standard reference work of the subject"[1] and likely to "become the standard reference for this generation. A complete treatment of the foundations, and the definitive exposition of the principal bundle point of view."[2] Also described as having "complete and well organized proofs of the numerous 'folk theorems,' the systematic and well thought out treatment of the theory of connections in principal fibre bundles, and the closing notes giving a skillful survey of the wide horizons."[3] A review of Volume II says "The book contains much material not otherwise available in book form and sometimes not easily accessible."[4]
It is weak on geometric intuition. Reviews describe it as having "austere abstraction",[5] and some of the chapters are said "to be part of a conspiracy to blindfold [the readers] geometric vision."[5] Another reviewer says "not the slightest attempt is made to present to the modern reader the marvelous geometric insights into Riemannian geometry pioneered by Cartan".[3]
It is not suitable as an introduction. Reviews say it has "no exercises, almost no examples"[3]. Also that "the reader is expected to come equipped with some earlier familiarity"[5] with differential geometry and "The first chapter is a very sketchy introduction to the background material in manifolds, tensors, Lie groups, Lie algebras, and fiber bundles."[6]. One reviewer says "anyone who thrusts these two volumes on a beginner for use as an introductory text would be guilty of committing an act of inhumanity against a fellow being.[1]
