S. Alexander, V. Kapovitch, A. Petrunin: “Alexandrov Geometry”. AMS, 2024, xviii + 282 pp

Alexandrov geometry is an elementarily accessible theory of curvature, based on the geometry of triangles. It is a close relative of classical Euclidean geometry from its pre-Cartesian era. This theory captures the essence of sectional curvature, one of the central concepts in Riemannian geometry, and enables the study of non-smooth spaces that cannot be analyzed using differential geometry.

While historically incorrect, one might see the ideal origin of Alexandrov geometry in the following two important infinitesimal-to-local-to-global results in Riemannian geometry. Firstly, the Cartan–Hadamard theorem, restated in the following way:

Secondly, Toponogov’s Theorem, restated as follows:

The concepts of a Riemannian manifold with non-positive or non-negative sectional curvature are not particularly intuitive and require a foundational understanding of differential geometry, including smooth manifolds, tangent bundles, Riemannian metrics, connections, Riemannian tensors, and more. In contrast, the equivalent properties postulated by the two theorems above seem much simpler and more accessible: the concepts of triangles and angles are familiar to every middle school student.

As it turns out, the condition on the sum of the angles can be replaced by another condition that only involves the distances from a vertex of the triangle to the midpoint of the opposite side.

For any 3 points \(A\), \(B\), \(C\) in the complete, smooth Riemannian manifold \(X\), connect the points by geodesics (i.e., curves of shortest length) \(a\), \(b\), \(c\) and denote by \(M\) the midpoint of the geodesic \(a\) connecting \(B\) and \(C\). We consider now the comparison triangle \(\tilde{A} \tilde{B} \tilde{C} \subset \mathbb{R}^{2}\) of \(ABC\) which has the same side lengths as the triangle \(ABC\). Note that the comparison triangle is uniquely defined up to a rigid motion of the Euclidean plane and let \(\tilde{M}\) be the midpoint of the segment \(\tilde{B}\tilde{C}\).

There is no reason for the distance between \(d_{X}(A,M)\) between \(A\) and \(M\) in the manifold \(X\) to coincide with the Euclidean distance \(d_{\mathbb{R}^{2}}(\tilde{A}, \tilde{M})\). Indeed, the equality valid for all triangles in the Riemannian manifold \(X\) is equivalent to the statement that \(X\) is itself a Euclidean space.

On the other hand, by a version of the Cartan–Hadamard theorem, the manifold \(X\) is simply connected and has non-positive curvature if and only if for all triangles \(ABC\) in \(X\).

Similarly, by a version of the Toponogov theorem, the manifold \(X\) has non-negative sectional curvature if and only if for all triangles \(ABC\) in \(X\).

In other words, non-positive sectional curvature (plus simply connectedness) is equivalent to having all triangles thin, while non-negative curvature is equivalent to having all triangles thick, as described in equations (0.2) and (0.3), respectively.

From the perspective of classical Riemannian geometry, the Cartan–Hadamard and Toponogov theorems provide geometric tools for investigating important classes of Riemannian manifolds, including all irreducible symmetric spaces. Alexandrov geometry takes this a step further by considering the comparison statements (0.2) and (0.3) as definitions of non-positive and non-negative curvature, respectively. An important advantage of this approach, beyond its emphasis of geometric over analytic tools, is that it requires much fewer prerequisites, both from the scholar and, more importantly, from the space in question. Specifically, both definitions are fully meaningful for all metric spaces in which any pair of points is connected by a geodesic – a curve whose length equals the distance between its endpoints.

This perspective allows a unified treatment of spaces that are impossible to approach using classical differential geometry. The class of non-negatively curved metric spaces includes all non-smooth, possibly polyhedral boundaries of convex bodies in Euclidean spaces; all spaces of orbits of isometric group actions on Euclidean spaces; the Wasserstein space of measures on Euclidean spaces. On the other hand, the class of non-positively curved metric spaces, as defined above, includes all metric trees, affine buildings, certain spaces of maps into Riemannian manifolds important in the theory of harmonic maps, the Teichmüller space and many polyhedral spaces playing an important role in geometric group theory.

The strength of Alexandrov geometry lies in its generality and applicability to a broad variety of situations. Beyond classical Riemannian geometry it has applications to geometric group theory, Teichmüller theory, harmonic map theory, superrigidity, topological stability, theory of algebraic, arithmetic and Cremona groups, topological finiteness, metric measure theory, the theory of transformation groups and geometrization of three-manifolds. Alexandrov geometry has also paved the way, both in terms of the methods and ideas, for the synthetic theory of spaces with lower bounds on the Ricci curvature flourishing in the last two decades.

The theory has an intrinsic beauty and resembles in many respects classical Euclidean geometry with its study of configurations of basic geometric figures by using very intuitive arguments. While some advanced arguments of the theory use analytic tools like gradient flows of concave functions, the whole theory can be accessed and appreciated by any student with a minimal background in analysis and interest in geometry. However, for a better appreciation, some knowledge of Riemannian geometry is recommended.

Two central results of Alexandrov geometry are the following local-to-global versions of the Cartan–Hadamard and the Toponogov theorems, valid in the general context of geodesic metric spaces:

Both local-to-global theorems have strong topological and geometric applications. For instance, a globally non-positively curved metric space is contractible and has a convex distance function. On the other hand, any non-negatively curved locally compact metric space is homotopy equivalent to a compact convex subset, whose sum of the Betti numbers is bounded in terms of the dimension. The classes of spaces with upper/lower curvature bounds behave well under many natural operations with metric spaces. More importantly, these classes are stable with respect to Gromov–Hausdorff convergence, and even more general types of convergence. Thus, understanding singular spaces with curvature bounds allows to draw conclusions in classical Riemannian geometry. A typical deep theorem using limit arguments of this type and structural features of limit spaces is the statement that on a Riemannian manifold of a fixed dimension and a lower curvature bound, the curvature tensor has uniformly bounded integral on a ball of a fixed radius.

Alexandrov geometry, developed in the mid-20th century, has gained tremendous momentum through the works of Gromov and has flourished over the past 30-40 years. The theory of non-positively curved spaces played a central role in the geometric group theory community. There are several very good introductions to the theory of non-positively curved spaces, with a focus on applications to geometric group theory. However, sources for the basics of Alexandrov geometry with lower curvature bounds are scarce. Often, scholars interested in basic problems of the theory need to read the original papers, which can be difficult to digest.

The book by Alexander–Kapovitch–Petrunin closes this fundamental gap. It develops the basics of the theories of spaces with upper and lower curvature bounds from scratch, emphasizing the similarity between the two theories. The book aims to achieve two goals: to serve as a foundational source for the theory, presenting most results in their most general form, and to require essentially no prerequisites, making it suitable for graduate students eager to learn an elementary but deep geometric theory. It can also serve as a base or background for a graduate course. For the foundations of the theory of lower curvature bounds this book will undoubtedly be an exteremely valuable resource, as most of the material is only available in the original papers, with some content being completely new. But also for the theory of spaces with upper curvature bounds, the book offers fresh perspectives and contains material for the first time outside the original research papers, such as Reshetnyak’s majorization theorem, the theorem of Kirzsbraun–Lang–Schroeder and Kleiner’s dimension theory.

A single book of a reasonable size certainly cannot encompass all material on Alexandrov geometry, so the authors had to make certain choices. In particular, they decided not to address isometric actions on spaces with curvature bounds, nor to include specific results about some special types of Alexandrov spaces like boundaries of convex sets or affine buildings. The book covers all fundamental concepts of Alexandrov geometry, which do not rely on finite-dimensionality of the spaces and are not related to isometric group actions. Hopefully, a continuation of the book covering the theory of finite-dimensional Alexandrov spaces will appear at some point in the future.

A few more details: the book includes a concise summary of general metric geometry, the basics of the theory of spaces with upper and with lower curvature bounds, including globalization results, the majorization theorem of Reshetnyak, the theory of dimension for Alexandrov spaces and the theory of gradient flows on such spaces.

The most general form of the theory chosen by the authors sometimes makes the reading more challenging. However, the proofs are concise, and their core ideas are clearly presented. The notations used in the book are intuitive, though somewhat unconventional. While this choice is well-suited for in-depth reading, it can make quick reference more difficult. Despite this minor drawback, I am very happy that this book has finally been published after many years of preparation. I am confident that it will be a valuable resource for mathematicians interested in using Alexandrov geometry and that it will contribute to the broader dissemination and popularization of this beautiful geometric theory.