Foundations of Differential Geometry (Wiley Classics Library) (Volume 2)

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Both were published again in as Wiley Classics Library. The first volume considers manifolds , fiber bundles , tensor analysis , connections in bundles , and the role of Lie groups. It also covers holonomy , the de Rham decomposition theorem and the Hopf—Rinow theorem.

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According to the review of James Eells , it has a "fine expositional style" and consists of a "special blend of algebraic, analytic, and geometric concepts". Eells says it is "essentially a textbook even though there are no exercises ".

An advanced text, it has a "pace geared to a [one] term graduate course". The second volume considers submanifolds of Riemannian manifolds , the Gauss map , and the second fundamental form. It continues with geodesics on Riemannian manifolds, Jacobi fields , the Morse index , the Rauch comparison theorems , and the Cartan—Hadamard theorem. In a discussion of curvature representation of characteristic classes of principal bundles Chern—Weil theory , it covers Euler classes , Chern classes , and Pontryagin classes.

The second volume also received a favorable review by J. Eells in Mathematical Reviews.

Foundations of Differential Geometry, 2 Volume Set

After obtaining his mathematics degree from the University of Tokyo and his Ph. Kobayashi's research spans the areas of differential geometry of real and complex variables, and his numerous resulting publications include several book: Foundations of Differential Geometry with N. Shoshichi Kobayashi , Katsumi Nomizu. This two-volume introduction to differential geometry, part of Wiley's popular Classics Library, lays the foundation for understanding an area of study that has become vital to contemporary mathematics.

Differential Geometry - Math History - NJ Wildberger

The second volume continues with the study of variational problems on geodesics through differential geometric aspects of characteristic classes. Both volumes familiarize readers with basic computational techniques.