##
**Geometry of nonpositively curved manifolds.**
*(English)*
Zbl 0883.53003

Chicago Lectures in Mathematics. Chicago, IL: The University of Chicago Press. 449 p. (1996).

The monograph under review is devoted to the geometry of manifolds of nonpositive sectional curvature and modern developments around the Mostow rigidity theorem for symmetric spaces of noncompact type and rank at least 2. The book is based on a series of lectures held in Tokyo in 1985 and a number of relevant research articles by the author. It represents a different approach to the subject than [W. Ballmann, M. Gromov and V. Schroeder, ‘Manifolds of nonpositive curvature’ (Progress in Math. 61, Birkhäuser, Boston) (1985; Zbl 0591.53001), referred to as [BGS] in the following].

The first chapter introduces to the theory of complete connected manifolds of nonpositive sectional curvature and in particular of the simply connected ones. The latter spaces are diffeomorphic to Euclidean spaces and show similar convexity properties. A very important tool in the study of simply connected nonpositively curved manifolds \(\widetilde{M}\) is the construction of a boundary sphere \(\widetilde M(\infty)\) together with a natural topology such that the isometries of \(\widetilde{M}\) extend to homeomorphisms of \(\widetilde M(\infty)\). The action of the isometry group \(I(\widetilde{M})\) on \(\widetilde M(\infty)\) yields important information about geometrical properties of the manifold \(\widetilde{M}\) itself and its quotients, especially in the case when \(I(\widetilde{M})\) is “large”, satisfying the duality condition as defined in two different ways in Section 1.9. This condition is automatically satisfied if \(\widetilde M(\infty)\) is the universal covering space of a complete manifold \(M\) of finite volume.

Chapter 2 is devoted to the description of the geometric structure of symmetric spaces of noncompact type and rank at least 2. Details of the theory are illustrated for the standard example \(SL(n,{\mathbb{R}}) / SO(n,{\mathbb{R}})\).

In the first half of Chapter 3 the author studies the concept of the Tits metric Td on \(\widetilde M(\infty)\) as introduced by Gromov and discussed in detail in [BGS, Section 4]. Then, further results about the Tits geometry and the Tits partial ordering for symmetric spaces of noncompact type are investigated, especially those which are relevant for the proof of Mostow’s rigidity theorem. The chapter closes with a brief discussion of the boundary \(K\widetilde M(\infty)\) introduced by Karpelevič for symmetric spaces of noncompact type and rank at least 2.

Chapter 4 deals with the study of the action of isometries of a symmetric space \(\widetilde M(\infty)\) of noncompact type and rank at least 2 on the boundary sphere \(\widetilde M(\infty)\). Isometries \(\phi\) of \(\widetilde{M}\) can be classified into 3 types, the elliptic, parabolic, and axial ones, respectively, leading to different characterisations of the fixed point set \(\widetilde M_{\phi}(\infty)\subset\widetilde M(\infty)\) of \(\phi\). For example, in Section 4.2, the author presents two geometric versions of the Bruhat decomposition for \(G=I_0(\widetilde{M})\).

Chapter 5 contains a criterion for simply connected nonpositively curved Riemannian manifolds \(\widetilde M\) to split as a Riemannian product. In some cases splitting criteria can be formulated entirely in terms of the Tits metric Td on \(\widetilde M(\infty)\).

In Chapter 6, the author investigates properties of isometries of Euclidean space such as commutator estimates, and of solvable subgroups. For example, it is shown that a solvable subgroup \(\Gamma\) of \(I({\mathbb{R}}^n)\) leaves invariant some \(k-\)flat \(F\) in \({\mathbb{R}}^n\) such that \(F/\Gamma\) is compact. This result helps to derive a slight generalization of one of the Bieberbach theorems. All these considerations are useful in Chapter 7 where simply connected spaces are studied whose Euclidean de Rham factor is nontrivial.

Chapter 8 starts with a discussion of two equivalent statements of Mostow’s rigidity theorem. Making use of the geometry on \(\widetilde M(\infty)\) as discussed in Sections 3.7 up to 3.11, the author presents a modified proof of Mostow’s original one.

Chapter 9 contains a number of further analogues or generalizations of the Mostow rigidity theorem which yield characterizations of symmetric spaces of noncompact type in terms of the geometry of a complete simply connected manifold \(\widetilde{M}\) of nonpositive sectional curvature. Results of Gromov, Ballmann, and Burns-Spatzier are included.

Finally, in Chapter 10, algebraic properties of the fundamental group are related to geometric properties of a compact manifold \(M\) of nonpositive curvature. It contains a collection of various results. For example, the topology of \(M\) allows to conclude whether \(M\) splits as a Riemannian product (Gromoll-Wolf, Lawson-Yau), whether \(M\) is flat (Bieberbach, Yau, Zimmer, and others), or whether \(M\) is irreducible and locally symmetric of rank at least 2 (Ballmann-Eberlein).

This book is very nicely written and serves as an excellent source on nonpositively curved spaces. It is clearly structured with each of the 10 chapters preceded by a detailed table of contents. Proofs which are omitted are either presented in an appendix at the end of a chapter or are referred to the adequate places in the literature. The bibliography is exhaustive and followed by a helpful, 9 pages long index of notations.

The first chapter introduces to the theory of complete connected manifolds of nonpositive sectional curvature and in particular of the simply connected ones. The latter spaces are diffeomorphic to Euclidean spaces and show similar convexity properties. A very important tool in the study of simply connected nonpositively curved manifolds \(\widetilde{M}\) is the construction of a boundary sphere \(\widetilde M(\infty)\) together with a natural topology such that the isometries of \(\widetilde{M}\) extend to homeomorphisms of \(\widetilde M(\infty)\). The action of the isometry group \(I(\widetilde{M})\) on \(\widetilde M(\infty)\) yields important information about geometrical properties of the manifold \(\widetilde{M}\) itself and its quotients, especially in the case when \(I(\widetilde{M})\) is “large”, satisfying the duality condition as defined in two different ways in Section 1.9. This condition is automatically satisfied if \(\widetilde M(\infty)\) is the universal covering space of a complete manifold \(M\) of finite volume.

Chapter 2 is devoted to the description of the geometric structure of symmetric spaces of noncompact type and rank at least 2. Details of the theory are illustrated for the standard example \(SL(n,{\mathbb{R}}) / SO(n,{\mathbb{R}})\).

In the first half of Chapter 3 the author studies the concept of the Tits metric Td on \(\widetilde M(\infty)\) as introduced by Gromov and discussed in detail in [BGS, Section 4]. Then, further results about the Tits geometry and the Tits partial ordering for symmetric spaces of noncompact type are investigated, especially those which are relevant for the proof of Mostow’s rigidity theorem. The chapter closes with a brief discussion of the boundary \(K\widetilde M(\infty)\) introduced by Karpelevič for symmetric spaces of noncompact type and rank at least 2.

Chapter 4 deals with the study of the action of isometries of a symmetric space \(\widetilde M(\infty)\) of noncompact type and rank at least 2 on the boundary sphere \(\widetilde M(\infty)\). Isometries \(\phi\) of \(\widetilde{M}\) can be classified into 3 types, the elliptic, parabolic, and axial ones, respectively, leading to different characterisations of the fixed point set \(\widetilde M_{\phi}(\infty)\subset\widetilde M(\infty)\) of \(\phi\). For example, in Section 4.2, the author presents two geometric versions of the Bruhat decomposition for \(G=I_0(\widetilde{M})\).

Chapter 5 contains a criterion for simply connected nonpositively curved Riemannian manifolds \(\widetilde M\) to split as a Riemannian product. In some cases splitting criteria can be formulated entirely in terms of the Tits metric Td on \(\widetilde M(\infty)\).

In Chapter 6, the author investigates properties of isometries of Euclidean space such as commutator estimates, and of solvable subgroups. For example, it is shown that a solvable subgroup \(\Gamma\) of \(I({\mathbb{R}}^n)\) leaves invariant some \(k-\)flat \(F\) in \({\mathbb{R}}^n\) such that \(F/\Gamma\) is compact. This result helps to derive a slight generalization of one of the Bieberbach theorems. All these considerations are useful in Chapter 7 where simply connected spaces are studied whose Euclidean de Rham factor is nontrivial.

Chapter 8 starts with a discussion of two equivalent statements of Mostow’s rigidity theorem. Making use of the geometry on \(\widetilde M(\infty)\) as discussed in Sections 3.7 up to 3.11, the author presents a modified proof of Mostow’s original one.

Chapter 9 contains a number of further analogues or generalizations of the Mostow rigidity theorem which yield characterizations of symmetric spaces of noncompact type in terms of the geometry of a complete simply connected manifold \(\widetilde{M}\) of nonpositive sectional curvature. Results of Gromov, Ballmann, and Burns-Spatzier are included.

Finally, in Chapter 10, algebraic properties of the fundamental group are related to geometric properties of a compact manifold \(M\) of nonpositive curvature. It contains a collection of various results. For example, the topology of \(M\) allows to conclude whether \(M\) splits as a Riemannian product (Gromoll-Wolf, Lawson-Yau), whether \(M\) is flat (Bieberbach, Yau, Zimmer, and others), or whether \(M\) is irreducible and locally symmetric of rank at least 2 (Ballmann-Eberlein).

This book is very nicely written and serves as an excellent source on nonpositively curved spaces. It is clearly structured with each of the 10 chapters preceded by a detailed table of contents. Proofs which are omitted are either presented in an appendix at the end of a chapter or are referred to the adequate places in the literature. The bibliography is exhaustive and followed by a helpful, 9 pages long index of notations.

Reviewer: Ruth Kellerhals (Göttingen)

### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53C20 | Global Riemannian geometry, including pinching |

53C35 | Differential geometry of symmetric spaces |