This book gives an introduction into the theory of metric spaces of non-positive curvature in the sense of H. Busemann. The author begins with the description of the fundamental works by Hadamard on surfaces of non-positive curvature and gives a short survey of foundational works in metric geometry by K. Menger, A. Wald, H. Busemann and A. D. Aleksandrov.
Chapters 1 and 2, deal with elementary preliminary topics such as metric spaces, length, length spaces (otherwise known as metric spaces with intrinsic metric, or inner metric spaces), geodesics, geodesic spaces (also known as geodesically connected metric spaces) geodesic convexity and Menger’s convexity.
Chapter 3 presents a preliminary material on Lipschitz and, in particular, non-expanding mappings, distance non-decreasing mappings, local isometries and covering spaces.
In Chapter 4, the author presents familiar Hausdorff and Busemann-Hausdorff distances and introduces different metrics on isometry groups of metric spaces.
Chapters 5 and 6 deal with convexity and convex functions in different vector spaces.
In Chapter 7, the author presents different criteria for uniqueness of minimal geodesics in normed vector spaces. In particular, it is shown that in a normed vector space, minimal geodesics are unique if and only if the normed vector space is strictly convex. If \(\gamma \left( s\right) \), \( a\leq s\leq b\), is an arc length parametrization of a non-constant geodesic \( \mathcal{L}\), then \(\gamma \left( \left( \left( d-c\right) t+\left( bc-ad\right) \right) /\left( b-a\right) \right) \), \(c\leq t\leq d\), is called an affinely parametrized geodesic.
Chapter 8 begins with the following definition of Busemann spaces: A geodesic metric space \(\left( X,\rho \right) \) is called a Busemann space if for any two affinely parametrized geodesics \(\gamma :\left[ a,b\right] \rightarrow X\), \(\gamma ^{\prime }:\left[ a^{\prime },b^{\prime }\right] \rightarrow X\), the function \(D_{\gamma ,\gamma ^{\prime }}\left( t,t^{\prime }\right) =\rho \left( \gamma \left( t\right) ,\gamma \left( t^{\prime }\right) \right) ,\) \( t\in \left[ a,b\right] ,\) \(t^{\prime }\in \left[ a^{\prime },b^{\prime } \right] ,\) is convex. The author proves Proposition 8.1.2 which provides a number of equivalent definitions of Busemann spaces, among which the reader can find Busemann’s original definition of non-positive curvature: If \( \mathcal{T=}abc\) is a geodesic triangle in a geodesic space \(\left( X,\rho \right) \) and \(m_{1},m_{2}\) are the midpoints of the sides \(ab\) and \(ac\), then \(\rho \left( m_{1},m_{2}\right) \leq \rho \left( b,c\right) /2\). The examples of Busemann spaces provided in Section 8.1 are also examples of Aleksandrov \(\operatorname{Re} _{0}\) domains (otherwise known as \(\text{CAT}\left( 0\right) \) domains) except for Proposition 8.1.6: Each strictly convex normed space is a Busemann space (in contrast, by a theorem of A. D. Aleksandrov, a normed vector space is of Aleksandrov curvature \(\leq 0\) if and only if it is an inner product vector space). The author proves basic properties of Busemann spaces: uniqueness of geodesics, each Busemann space is contractible, every (local) geodesic is minimal geodesic. The remainder of Chapter 8 deals with the geodesic convexity and convex functions in Busemann spaces.
In Chapter 9, the author presents the following theorem by S. B. Alexander and R. L. Bishop: In a complete locally compact and locally convex geodesic metric space (i.e., locally Busemann space) \(\left( X,\rho \right) \), a naturally defined mapping \(\exp _{x}\) is a universal covering map of \(X\), and also a theorem by M. Gromov: Every complete geodesic locally compact, locally convex and simply connected metric space is a Busemann space. Two geodesic rays \(r_{1},r_{2}:[0,+\infty )\rightarrow X\) in a metric space \(\left( X,\rho \right) \) are called asymptotic if there is \(\alpha >0\) such that \(\left| r_{1}\left( t\right) -r_{2}\left( t\right) \right| \leq \alpha \) for every \(t\geq 0\). Let \(p\in X\) and let \(R_{p}\) denote the set of all geodesic rays emanating from \(p\), equipped with the topology of uniform convergence on compact sets. Two rays starting at \(p\) are called equivalent if they are asymptotic. The quotient space of \(R_{p}\) w.r.t. the equivalence relation just introduced is called the visual boundary \(\partial _{p}X\).
Chapter 10, deals with the visual boundary of a metric space, some of its properties and examples. The two remaining chapters are more specialized.
In Chapter 11, the author presents the results related to minimal displacement (i.e., \(\lambda \left( f\right) =\inf_{x\in X}\rho \left( x,f\left( x\right) \right) \)) and the minimal set (i.e., \(\text{Min} \left( f\right) =\left\{ x\in X\text{ : }\rho \left( x,f\left( x\right) \right) =\lambda \left( f\right) \right\} \)) of a map \(f:X\rightarrow X\), and studies the related classification of isometries of the metric space \( \left( X,\rho \right) \). These concepts are applied to the study of axial isometries, closed geodesics and parallel geodesics. Let \(r\left( t\right) \) , \(t\geq 0\), be a geodesic ray in a metric space \(\left( X,\rho \right) \). Then the associated (convex) Busemann function is defined by \(B_{r}\left( x\right) =\lim_{t\rightarrow +\infty }\left[ \rho \left( x,r\left( t\right) \right) -t\right] \); hence, the distance between \(x\) and \(r\left( t\right) \) is approximately equal to \(B_{r}\left( x\right) +t\) for large \(t\). A level set of a Busemann function is called a horosphere.
In Chapter 12, basic properties of Busemann functions are presented, a short introduction into Busemann’s theory of co-rays is given and relations between co-rays, Busemann functions and horospheres are discussed.
The book is well written and is easy to read. The first part of the book is somewhat elementary and should be accessible to seniors and first year graduate students. The second part of the book is more specialized and research-oriented, and could serve for researchers as a good source of information about metric geometry.