Foundations of Differential Geometry, Vol. 1

Shoshichi Kobayashi , Katsumi Nomizu, "Foundations of Differential Geometry, Vol. 1"
Wiley-Interscience; New Ed edition (February 8, 1996) | ISBN: 0471157333 | 329 pages | Djvu | 6,6 M
Review:

The Definitive Reference for Four Decades

The two-volume set by Kobayashi and Nomizu has remained the definitive reference for differential geometers since their appearance in 1963(volume 1) and 1969 (volume 2). Over the decades, many readers have developed a love/hate relationship with these difficult, challenging texts. For example, in a 2006 edition of a competing text, the author remarked that "every differential geometer must have a copy of these tomes," but followed this judgment by observing that "their effective usefulness had probably passed away," comparing them to the infamously difficult texts of Bourbaki.

As a practicing differential geometer, I would argue that Kobayashi and Nomizu remains an essential reference even today, for a number of reasons.
Volume 1 still remains unrivalled for its concise, mathematically rigorous presentation of the theory of connections on a principal fibre bundle---material that is absolutely essential to the reader who desires to understand gauge theories in modern physics. The essential core of Volume 1 is the development of connections on a principal fibre bundle, linear and affine connections, and the special case of Riemannian connections, where a connection must be "fitted" to the geometry that results from a pre-existing metric tensor on the underlying manifold, M.
Volume 2 offers thorough introductions to a number of classical topics, including submanifold theory, Morse index theory, homogeneous and symmetric spaces, characteristic classes, and complex manifolds.

The influence of the texts by Kobayashi and Nomizu can be seen in most of the subsequent differential geometry texts, both in organization and content, and especially in the adoption of notation. If there was a particularly fine point in your favorite introductory differential geometry text that you never completely understood, the odds are good that you will find the answer, fully developed and presented at an entirely different mathematical level, in Kobayashi and Nomizu. It is not an unreasonable analogy to say that learning differential geometry without having your own copy of Kobayashi/Nomizu is like studying literature in the complete ignorance of Shakespeare.

Let there be no mistake about the advanced level of these texts. The Preface to Volume 1 clearly states that the authors presume the reader to be familiar with differentiable manifolds, Lie groups, and fibre bundles, as developed in the (now classical) texts by Chevalley, Montgomery-Zippin, Pontrjagin, and Steenrod. Today's reader is far more likely to have studied these subject from more recent books like those by Boothby, Hall, and Husemoller, but whatever the source, a familiarity IS presumed. The "lightning review" provided in Chapter I of Volume 1 will be extremely tough going for the reader who is new to these topics. It should also be noted that in 329 pages of Volume 1 and 470 pages of Volume 2, not a single diagram or picture is to be found! Those drawn to geometry for its visual aspects will find Kobayashi/Nomizu totally lacking in visual aids.

As with so many classic references in mathematics, the hardbound edition of Kobayashi and Nomizu is no longer in print. Copies appear sporadically on the used book market at absolutely obscene prices. The Classics Library paperback edition is still available, but the serious student will find that the paperbacks simply do not fare well under serious, sustained use.

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