First published in 1964, this book served as a text on differential geometry
to several generations of graduate students all over the world.
The first half of the book (Chapters 1–6) presents basics of
the theory of manifolds, vector bundles, differential forms, and Lie
groups, with a special emphasis on the theory of linear and affine
connections. The second half of the book (Chapters 7–11) is
devoted to Riemannian geometry. Following the definition and main
properties of Riemannian manifolds, the authors discuss the theory of
geodesics, complete Riemannian manifolds, and curvature. Next, they
introduce the theory of immersion of manifolds and the second
fundamental form. The concluding Chapter 11 contains more complicated
results on which much of the research in Riemannian geometry is based:
the Morse index theorem, Synge's theorem on closed geodesics, Rauch's
comparision theorem, and Bishop's volume-comparision theorem.
Clear, concise writing as well as many exercises and examples make this
classic an excellent text for a first-year graduate course on differential
geometry.
Readership
Graduate students and research mathematicians interested in
geometry and topology.