##
**Lectures on spaces of nonpositive curvature. With an appendix by Misha Brin: Ergodicity of geodesic flows.**
*(English)*
Zbl 0834.53003

DMV Seminar. Bd. 25. Basel: Birkhäuser Verlag. 112 p. (1995).

This book grew out of a series of lectures given by the author at a DMV seminar in Blaubeuren, Germany. Its avowed goal is to present an improved version of the proof of the rank rigidity theorem for manifolds of nonpositive curvature, which was proved for complete, finite volume manifolds of bounded nonpositive sectional curvature by the author and independently by Burns-Spatzier around 1984. The book achieves this and at the same time provides an introduction to the modern treatment of nonpositively curved spaces in which one does as much geometry as possible in the context of metric spaces with suitable properties, not in the context of smooth Riemannian manifolds. The extra generality and flexibility of this approach is extremely useful and tends to focus attention on the underlying geometric ideas that make things work and away from convenient but inessential properties, which in many cases include differentiability. Moreover, it is sometimes easier to obtain results for Riemannian manifolds with various curvature hypotheses by proving these results for a more general class of spaces. This way of doing geometry goes back many years to the work of A. D. Alexandrov, S. Cohn-Vossen und W. Rinow, but it has reemerged as a strong force in the last 15 years. In recent times M. Gromov provided the initial impetus for doing geometry in metric spaces, but this theme has been developed further by many geometers including the author.

In the study of spaces with nonpositive curvature one must also acknowledge the significant influence of H. Busemann, who was one of the first geometers to use synthetic and axiomatic methods to do geometry in spaces with no differentiable structure. Although the study of spaces of nonpositive curvature has evolved in a direction rather different than he proposed, it bears the clear marks of Busemann’s work, most notably in the notion of Busemann function.

In addition to the present book there are other recent books on nonpositively curved spaces and the related hyperbolic spaces, which are metric space generalizations due to Gromov of simply connected, complete Riemannian manifolds with sectional curvatures bounded above by a negative constant. These include works listed in the references by Ballmann, Gromov and Schroeder; Bridson and Haefliger; Gromov; Ghys and de la Harpe. A set of lecture notes by Eberlein will also be available soon from the University of Chicago Press.

In the present nondifferentiable approach to geometry the basic object of study is a metric space \((X,d)\), and one then requires further the properties of geodesic conn geodesic completeness and curvature bounds (upper or lower) as needed for applications. In chapter I of the present book the author discusses the interior geometry of a metric space, which is the framework for the current approach. The metric \(d\) of \(X\) determines an inner pseudometric \(d\), as follows:

For each continuous curve \(\gamma : [a,b] \to X\) define the length \(L(\gamma)=\sup \{\sum^N_{i=1} d (\gamma (t_i)\), \(\gamma (t_{i+1})): a=t_1 < t_2 < \ldots < t_{N+1}=b\) is a partition of \([a,b]\}\). The length of a curve \(\gamma\) may of course be infinite. One then defines for any two points \(x,y\) of \(X\) the pseudodistance \(d_i (x,y)=\inf \{L (\gamma) : \gamma\) is a continuous curve in \(X\) from \(x\) to \(y\}\). Note that \(d_i (x,y) \geq d(x,y)\) and \(d_i (x,y)\) may be infinite. If \(d_i=d\), then \((X, d)\) is called an inner metric space. A Riemannian manifold \((X,d)\) is an inner metric space.

A curve \(\gamma : [a,b] \to X\) is called a geodesic of speed \(c \geq 0\) if for every \(t \in (a,b)\) there exists \(\varepsilon > 0\) such that \(d (\gamma (t_1)\), \(\gamma (t_2))=c |t_1-t_2 |\) for all \(t_1, t_2 \in (t-\varepsilon,t+\varepsilon)\). In particular the length of \(\gamma\) on \((t-\varepsilon,t+\varepsilon)\) equals \(d(\gamma (t-\varepsilon)\), \(\gamma (t+\varepsilon))\) so geodesics locally minimize distance. The metric space \((X,d)\) is geodesic if there exists a minimizing geodesic between any two points and is geodesically complete if all geodesics \(\gamma : [a,b] \to X\) can be extended maximally and defined on \((- \infty, \infty)\). If \((X,d)\) is a locally compact inner metric space, then \(X\) is geodesic and local and global versions of the Hopf-Rinow theorem exist.

One may define upper and lower curvature bounds for a geodesic space \((X,d)\) by comparing geodesic triangles in \((X,d)\) with geodesic triangles in a space of constant curvature \(\kappa\), which is a sphere, Euclidean space or hyperbolic space depending on whether \(\kappa > 0\), \(\kappa=0\) or \(\kappa < 0\). Let \(\Delta\) be a (suitable small) geodesic triangle with sides of lengths \(a,b,c\) and opposite vertices \(A,B,C\), and let \(\overline\Delta\) be a triangle in a space of constant curvature \(\kappa\) with the same side lengths and corresponding vertices \(\overline A, \overline B\) and \(\overline C\). The vertices of \(\Delta\) and \(\overline \Delta\) determine a bijection \(x \to \overline x\) between points on the edges of \(\Delta\) and points on the edges of \(\overline\Delta\). One says that \((X,d)\) has curvature \(\leq \kappa\) (respectively \(\geq \kappa)\) if for all points \(x,y\) on the edges of \(\Delta\) one has the inequality \(d(x,y) \leq \overline d (\overline x, \overline y)\) (respectively \(d(x,y) \geq \overline d (\overline x, \overline y))\). These definitions agree with the same curvature inequalities in the case that \((X,d)\) is a Riemannian manifold. Spaces with curvature \(\leq \kappa\) are called \(\text{CAT} (\kappa)\)-spaces, and the focus of the book is on CAT(0)-spaces. The geometry of spaces with curvature \(\geq \kappa\) is also extremely rich, but quite different from that of the \(\text{CAT} (\kappa)\)-spaces.

A complete, simply connected CAT(0)-space \((X,d)\) is called a Hadamard space. In Chapter II the author generalizes many of the methods and results about Riemannian manifolds with sectional curvature \(K \leq 0\) to the case of Hadamard spaces \((X,d)\). To begin with, one may construct the usual boundary at infinity \(X (\infty)\) using either asymptotic geodesic rays or the more general (but in this case equivalent) approach of Gromov that uses Busemann functions, which also may be defined directly in terms of the metric \(d\) without reference to geodesics. The author also discusses other properties of nonpositively curved Riemannian manifolds that generalize to Hadamard spaces \((X,d)\) including the Tits metric and geometry of \(X (\infty)\) and the classification of isometries \(\varphi\) of \(X\) by means of the displacement function \(d_\varphi : p \to d (p, \varphi p)\) for \(p \in X\).

In Chapter III the author extends his own work on geodesic flows and the Dirichlet problem for harmonic functions in rank 1 Riemannian manifolds \((X,d)\) of nonpositive sectional curvature. The extensions are valid for pairs \((X, \Gamma)\), where \(X\) is a geodesically complete Hadamard space satisfying a weak hyperbolicity condition and \(\Gamma\) is a suitably large group of isometries of \(X\). The hyperbolicity condition is that some geodesic of \(X\) does not bound a flat half plane in \(X\). We omit a precise definition of the condition of largeness for \(\Gamma\), the duality condition. This condition is satisfied if \(X\) is a Riemannian manifold and \(\Gamma\) is a discrete group of isometries of \(X\) such that the quotient space \(X/ \Gamma\) is a smooth manifold with finite volume. In this context the no flat half plane condition is equivalent to \(X\) having rank 1; that is, some geodesic of \(X\) admits no perpendicular, parallel, nonzero Jacobi vector field.

In the concluding Chapter IV the author presents a simplified proof of the rank rigidity theorem: Theorem: Let \(X\) be a complete, simply connected, smooth Riemannian manifold with nonpositive sectional curvature and rank \(k \geq 2\). If \(X\) is irreducible and \(I(X)\) satisfies the duality condition, then \(X\) is isometric to a symmetric space of rank \(k\).

The formulation above was proved by Eberlein and Heber ([EbHe] in the references). The proof here is shorter and more accessible and incorporates features of the proof in [EbHe] and the author’s original proof for the case that \(X\) admits a quotient manifold of finite volume.

The appendix written by M. Brin will be a valuable reference for geometers with a limited knowledge of the methods of ergodic theory and dynamical systems. Brin presents a self-contained and much simplified version of Anosov’s proof that the geodesic flow in the unit tangent bundle of a compact Riemannian manifold with strictly negative curvature is ergodic. In fact, the geodesic flow satisfies much stronger properties in this case.

This book is a fine introduction to the modern theory of spaces with curvature \(\leq 0\) and is written by one of the leading researchers in this field. Its simplified treatment of the rank rigidity theorem, its useful appendix and its modest length make it an attractive addition to one’s library, whether one is an expert or simply a person who wants to know more about the recent developments in this area.

In the study of spaces with nonpositive curvature one must also acknowledge the significant influence of H. Busemann, who was one of the first geometers to use synthetic and axiomatic methods to do geometry in spaces with no differentiable structure. Although the study of spaces of nonpositive curvature has evolved in a direction rather different than he proposed, it bears the clear marks of Busemann’s work, most notably in the notion of Busemann function.

In addition to the present book there are other recent books on nonpositively curved spaces and the related hyperbolic spaces, which are metric space generalizations due to Gromov of simply connected, complete Riemannian manifolds with sectional curvatures bounded above by a negative constant. These include works listed in the references by Ballmann, Gromov and Schroeder; Bridson and Haefliger; Gromov; Ghys and de la Harpe. A set of lecture notes by Eberlein will also be available soon from the University of Chicago Press.

In the present nondifferentiable approach to geometry the basic object of study is a metric space \((X,d)\), and one then requires further the properties of geodesic conn geodesic completeness and curvature bounds (upper or lower) as needed for applications. In chapter I of the present book the author discusses the interior geometry of a metric space, which is the framework for the current approach. The metric \(d\) of \(X\) determines an inner pseudometric \(d\), as follows:

For each continuous curve \(\gamma : [a,b] \to X\) define the length \(L(\gamma)=\sup \{\sum^N_{i=1} d (\gamma (t_i)\), \(\gamma (t_{i+1})): a=t_1 < t_2 < \ldots < t_{N+1}=b\) is a partition of \([a,b]\}\). The length of a curve \(\gamma\) may of course be infinite. One then defines for any two points \(x,y\) of \(X\) the pseudodistance \(d_i (x,y)=\inf \{L (\gamma) : \gamma\) is a continuous curve in \(X\) from \(x\) to \(y\}\). Note that \(d_i (x,y) \geq d(x,y)\) and \(d_i (x,y)\) may be infinite. If \(d_i=d\), then \((X, d)\) is called an inner metric space. A Riemannian manifold \((X,d)\) is an inner metric space.

A curve \(\gamma : [a,b] \to X\) is called a geodesic of speed \(c \geq 0\) if for every \(t \in (a,b)\) there exists \(\varepsilon > 0\) such that \(d (\gamma (t_1)\), \(\gamma (t_2))=c |t_1-t_2 |\) for all \(t_1, t_2 \in (t-\varepsilon,t+\varepsilon)\). In particular the length of \(\gamma\) on \((t-\varepsilon,t+\varepsilon)\) equals \(d(\gamma (t-\varepsilon)\), \(\gamma (t+\varepsilon))\) so geodesics locally minimize distance. The metric space \((X,d)\) is geodesic if there exists a minimizing geodesic between any two points and is geodesically complete if all geodesics \(\gamma : [a,b] \to X\) can be extended maximally and defined on \((- \infty, \infty)\). If \((X,d)\) is a locally compact inner metric space, then \(X\) is geodesic and local and global versions of the Hopf-Rinow theorem exist.

One may define upper and lower curvature bounds for a geodesic space \((X,d)\) by comparing geodesic triangles in \((X,d)\) with geodesic triangles in a space of constant curvature \(\kappa\), which is a sphere, Euclidean space or hyperbolic space depending on whether \(\kappa > 0\), \(\kappa=0\) or \(\kappa < 0\). Let \(\Delta\) be a (suitable small) geodesic triangle with sides of lengths \(a,b,c\) and opposite vertices \(A,B,C\), and let \(\overline\Delta\) be a triangle in a space of constant curvature \(\kappa\) with the same side lengths and corresponding vertices \(\overline A, \overline B\) and \(\overline C\). The vertices of \(\Delta\) and \(\overline \Delta\) determine a bijection \(x \to \overline x\) between points on the edges of \(\Delta\) and points on the edges of \(\overline\Delta\). One says that \((X,d)\) has curvature \(\leq \kappa\) (respectively \(\geq \kappa)\) if for all points \(x,y\) on the edges of \(\Delta\) one has the inequality \(d(x,y) \leq \overline d (\overline x, \overline y)\) (respectively \(d(x,y) \geq \overline d (\overline x, \overline y))\). These definitions agree with the same curvature inequalities in the case that \((X,d)\) is a Riemannian manifold. Spaces with curvature \(\leq \kappa\) are called \(\text{CAT} (\kappa)\)-spaces, and the focus of the book is on CAT(0)-spaces. The geometry of spaces with curvature \(\geq \kappa\) is also extremely rich, but quite different from that of the \(\text{CAT} (\kappa)\)-spaces.

A complete, simply connected CAT(0)-space \((X,d)\) is called a Hadamard space. In Chapter II the author generalizes many of the methods and results about Riemannian manifolds with sectional curvature \(K \leq 0\) to the case of Hadamard spaces \((X,d)\). To begin with, one may construct the usual boundary at infinity \(X (\infty)\) using either asymptotic geodesic rays or the more general (but in this case equivalent) approach of Gromov that uses Busemann functions, which also may be defined directly in terms of the metric \(d\) without reference to geodesics. The author also discusses other properties of nonpositively curved Riemannian manifolds that generalize to Hadamard spaces \((X,d)\) including the Tits metric and geometry of \(X (\infty)\) and the classification of isometries \(\varphi\) of \(X\) by means of the displacement function \(d_\varphi : p \to d (p, \varphi p)\) for \(p \in X\).

In Chapter III the author extends his own work on geodesic flows and the Dirichlet problem for harmonic functions in rank 1 Riemannian manifolds \((X,d)\) of nonpositive sectional curvature. The extensions are valid for pairs \((X, \Gamma)\), where \(X\) is a geodesically complete Hadamard space satisfying a weak hyperbolicity condition and \(\Gamma\) is a suitably large group of isometries of \(X\). The hyperbolicity condition is that some geodesic of \(X\) does not bound a flat half plane in \(X\). We omit a precise definition of the condition of largeness for \(\Gamma\), the duality condition. This condition is satisfied if \(X\) is a Riemannian manifold and \(\Gamma\) is a discrete group of isometries of \(X\) such that the quotient space \(X/ \Gamma\) is a smooth manifold with finite volume. In this context the no flat half plane condition is equivalent to \(X\) having rank 1; that is, some geodesic of \(X\) admits no perpendicular, parallel, nonzero Jacobi vector field.

In the concluding Chapter IV the author presents a simplified proof of the rank rigidity theorem: Theorem: Let \(X\) be a complete, simply connected, smooth Riemannian manifold with nonpositive sectional curvature and rank \(k \geq 2\). If \(X\) is irreducible and \(I(X)\) satisfies the duality condition, then \(X\) is isometric to a symmetric space of rank \(k\).

The formulation above was proved by Eberlein and Heber ([EbHe] in the references). The proof here is shorter and more accessible and incorporates features of the proof in [EbHe] and the author’s original proof for the case that \(X\) admits a quotient manifold of finite volume.

The appendix written by M. Brin will be a valuable reference for geometers with a limited knowledge of the methods of ergodic theory and dynamical systems. Brin presents a self-contained and much simplified version of Anosov’s proof that the geodesic flow in the unit tangent bundle of a compact Riemannian manifold with strictly negative curvature is ergodic. In fact, the geodesic flow satisfies much stronger properties in this case.

This book is a fine introduction to the modern theory of spaces with curvature \(\leq 0\) and is written by one of the leading researchers in this field. Its simplified treatment of the rank rigidity theorem, its useful appendix and its modest length make it an attractive addition to one’s library, whether one is an expert or simply a person who wants to know more about the recent developments in this area.

Reviewer: P.Eberlein (Chapel Hill)

### MSC:

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

53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |

53C70 | Direct methods (\(G\)-spaces of Busemann, etc.) |

37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |

53D25 | Geodesic flows in symplectic geometry and contact geometry |