Metric Structures in Differential Geometry
This text is aimed to be an elementary introduction to differential geometry. It consists of six Chapters. In Chapter 1 (“Differentiable manifolds”) the author treats, besides the standard results and concepts about manifolds, multilinear algebra and tensor fields as well as the de Rham cohomology and Stokes theorem. Chapter 2 is devoted to fiber bundles and is culminated by fundamental theorems about Grassmannians as universal bundles. Chapter 3 deals with homotopy groups and bundles over spheres. Chapter 4 is devoted to the theory of connections and curvature on manifolds and principal vector bundles. Chapter 5 has the title “Metric structures”. It deals with Riemannian connections, Riemannian curvature and related curvatures, isometric immersions and Riemannian submersions, variational calculus connected with minimizing properties of geodesics, theorems by Hadamard-Cartan and Bonnet-Myers, and finally, with actions of compact Lie groups on Riemannian manifolds. The most advanced chapter is Chapter 6, which discusses characteristic classes. It involves all of the most fundamental concepts and results, which one should expect from an introductory text. The style is rather concise and many facts are shifted to 165 nontrivial exercises. The book is very well written and can be recommended to those who want to learn the topic quickly and actively. I have only three comments: i) The title of the book is a bit misleading. One would expect under this title something like the reference , i.e., the direction developed by Gromov and his collaborators. ii) The (subject) index is not complete enough. iii) The bibliography (which is just a list of text-books about differential geometry) is sometimes not written carefully. For example, the book  by Helgason was published in 1978, not in 1962. The reference  to Kobayashi and Nomizu is referring only to the first volume, not to the second volume which appeared in 1969.