##
**Riemannian geometry.**
*(English)*
Zbl 0636.53001

Universitext. Berlin etc.: Springer-Verlag. xi, 248 p. (1987).

This book is an excellent textbook for first year graduate students, and provides the fundamental knowledge for anyone who wishes to specialize in pure mathematics or theoretical physics. Since the fifties of this century, Riemannian geometry has undergone drastic change in its contents. Instead of the local behavior treated by classical calculus, the interest of research in this discipline has shifted to the global feature of manifolds in connection with admissible Riemannian metrics. As a result, homology and homotopy theory for manifolds, de Rham’s cohomology theory, and Hodge’s decomposition theorem play an important rôle, and their relations with the Riemannian metric become indispensable in exposition. In view of this situation, the material compiled in this textbook may be considered appropriate, at least, in the opinion of the reviewer. Although the proofs of certain fundamental theorems, e.g., de Rham’s isomorphism theorem, have been omitted, they can be found easily in available treatises like F. W. Warner’s Foundations of differentiable manifolds and Lie groups (1983; Zbl 0516.58001) or S. Kobayashi and K. Nomizu’s Foundations of differential geometry, Vol. I (1963; Zbl 0119.37502), Vol. II (1969; Zbl 0175.48504).

The contents of this book is divided into five chapters: The first chapter is of preparatory character, presenting the fundamental notions of manifolds and related calculus. Here the reader can find the definitions and the fundamental properties of manifolds, submanifolds, smooth maps, tangent bundles, vector fields, covering maps and fibrations, tensors, and exterior forms.

In the second chapter the existence theorem of a Riemannian metric on a given manifold is proved thoroughly, then the author introduces the notions of covariant derivation, geodesics, completeness of manifolds. In this context the classical Hopf-Rinow theorem is proved. By the way, both the radius of injectivity and the cut locus are treated carefully. Moreover, the cut loci of spheres, real projective spaces, complex projective spaces, hyperbolic spaces, and flat tori are studied in detail.

Chapter III forms the central part of this book. Within this chapter, the author introduces the curvature tensor, Ricci curvature, and sectional curvature. Then the first and the second variations of the arc-length of curves are analyzed, and Jacobi vector fields are introduced. In order to study the geometry of homogeneous manifolds, the author carefully considers Riemannian submersions, and compares the sectional curvatures related by a Riemannian submersion \((\tilde M,\tilde g)\to (M,g)\). Finally, he classifies the various space forms, and proves the Hadamard- Cartan theorem concerning the Riemannian manifolds of nonpositive sectional curvature. We would like to mention that in section H of this chapter, the author exhibits the highly interesting results recently obtained by Gromov and Cheeger. Although the information given here seem to be sufficiently precise, yet the reader can find the detailed proof in the original papers pointed out in this context.

Chapter IV is devoted to the study of spectral geometry which is of current interest. In addition to the fundamental properties of the eigenvalues and of the eigenfunctions of the Beltrami-Laplace operator, the author also gives estimations of the eigenvalues due to Lichnerowicz, Obata, and Cheeger. With respect to the contents of this chapter, the monographs of M. Berger, P. Gauduchon and E. Mazet [Le spectre d’une variété riemannienne. (Springer Lect. Notes Math. 194) (1971; Zbl 0223.53034)], and of P. H. Bérard [Spectral geometry: direct and inverse problems. (Springer Lect. Notes Math. 1207) (1986; Zbl 0608.58001)] are helpful as references.

The last Chapter V gives a brief survey of Riemannian submanifolds, containing the celebrated Hadamard convexity theorem. In summary, after reading this textbook, the students would be convinced that Riemannian geometry remains to be a fruitful orchard waiting for cultivation.

The contents of this book is divided into five chapters: The first chapter is of preparatory character, presenting the fundamental notions of manifolds and related calculus. Here the reader can find the definitions and the fundamental properties of manifolds, submanifolds, smooth maps, tangent bundles, vector fields, covering maps and fibrations, tensors, and exterior forms.

In the second chapter the existence theorem of a Riemannian metric on a given manifold is proved thoroughly, then the author introduces the notions of covariant derivation, geodesics, completeness of manifolds. In this context the classical Hopf-Rinow theorem is proved. By the way, both the radius of injectivity and the cut locus are treated carefully. Moreover, the cut loci of spheres, real projective spaces, complex projective spaces, hyperbolic spaces, and flat tori are studied in detail.

Chapter III forms the central part of this book. Within this chapter, the author introduces the curvature tensor, Ricci curvature, and sectional curvature. Then the first and the second variations of the arc-length of curves are analyzed, and Jacobi vector fields are introduced. In order to study the geometry of homogeneous manifolds, the author carefully considers Riemannian submersions, and compares the sectional curvatures related by a Riemannian submersion \((\tilde M,\tilde g)\to (M,g)\). Finally, he classifies the various space forms, and proves the Hadamard- Cartan theorem concerning the Riemannian manifolds of nonpositive sectional curvature. We would like to mention that in section H of this chapter, the author exhibits the highly interesting results recently obtained by Gromov and Cheeger. Although the information given here seem to be sufficiently precise, yet the reader can find the detailed proof in the original papers pointed out in this context.

Chapter IV is devoted to the study of spectral geometry which is of current interest. In addition to the fundamental properties of the eigenvalues and of the eigenfunctions of the Beltrami-Laplace operator, the author also gives estimations of the eigenvalues due to Lichnerowicz, Obata, and Cheeger. With respect to the contents of this chapter, the monographs of M. Berger, P. Gauduchon and E. Mazet [Le spectre d’une variété riemannienne. (Springer Lect. Notes Math. 194) (1971; Zbl 0223.53034)], and of P. H. Bérard [Spectral geometry: direct and inverse problems. (Springer Lect. Notes Math. 1207) (1986; Zbl 0608.58001)] are helpful as references.

The last Chapter V gives a brief survey of Riemannian submanifolds, containing the celebrated Hadamard convexity theorem. In summary, after reading this textbook, the students would be convinced that Riemannian geometry remains to be a fruitful orchard waiting for cultivation.

Reviewer: Cheng-Chung Hwang (Nanking)

### MSC:

53-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry |

53B20 | Local Riemannian geometry |

53C20 | Global Riemannian geometry, including pinching |

58-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to global analysis |

58J60 | Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) |