Alexandrov geometry: foundations (1903.08539v6)

Abstract: Alexandrov spaces are defined via axioms similar to those given by Euclid. The Alexandrov axioms replace certain equalities with inequalities. Depending on the signs of the inequalities, we obtain Alexandrov spaces with curvature bounded above and curvature bounded below. The definitions of the two classes of spaces are similar, but their properties and known applications are quite different. Our approach is novel in its attention to the interrelatedness of the two fields, and its emphasis on the way each illuminates the other. The goal of this book is to give a comprehensive exposition of the structure theory of Alexandrov spaces with curvature bounded above and below. It includes all the basic material as well as selected topics inspired by considering the two contexts simultaneously. We only consider the intrinsic theory, leaving applications aside. This book includes material up to the definition of dimension. Another volume still in preparation will cover further topics.

• The paper establishes foundational definitions and key insights into Alexandrov spaces characterized by curvature inequalities.
• It demonstrates construction techniques such as Alexandrov's doubling theorem and Reshetnyak gluing to preserve curvature conditions.
• It reveals detailed analyses of geodesic behavior and dimensional theory while linking CAT(0) and Alex(0) spaces to broader geometric contexts.

An Essay on "Alexandrov Geometry: Foundations"

The paper "Alexandrov Geometry: Foundations," authored by Stephanie Alexander, Vitali Kapovitch, and Anton Petrunin, presents a comprehensive exploration of Alexandrov spaces, which arose from the paper of curvature properties in geometric contexts. These spaces are distinguished by curvature bounds described through inequalities rather than equalities, leading to the consideration of spaces characterized by either curvature bounded above (CBA) or below (CBB).

Alexandrov Spaces: Definitions and Comparisons

Alexandrov spaces are described via axiomatic frameworks that naturally extend traditional Euclidean geometrical concepts by replacing equalities with inequalities. This generalization gives rise to two primary classes of spaces: those with curvature bounded above (CBA) and those with curvature bounded below (CBB), each with distinct properties and implications.

The CBA spaces are often referred to as $\CAT(\kappa)$ spaces, named in honor of Cartan, Alexandrov, and Toponogov, and are characterized by their properties of uniqueness and non-branching nature of geodesics for points sufficiently close to each other. This is encapsulated by the requirement that any triangle within these spaces exhibits the so-called "thin triangle" property when compared with a reference space of constant curvature, $\Lob(2,\kappa)$.

Conversely, CBB spaces—typically referred to as $\Alex(\kappa)$ spaces—are distinguished by their robustness in maintaining separation or distance between points across the space, effectively dictating that any quadrilateral within the space should feature upper triangle inequality properties across its diagonals.

Key Results and Construction Techniques

The authors delve deeply into foundational results central to Alexandrov geometry. This includes substantial discussion on geodesic behavior, such as the existence and extension of geodesics in $\CAT(\kappa)$ spaces due to the triangle comparison theorem, and G-delta geodesicity in $\Alex(\kappa)$ spaces, confirming that most points have unique minimizing geodesics extending from any given start point.

The book also revisits historical constructions, such as Alexandrov's doubling theorem, which facilitates constructing new spaces by gluing along the boundaries of existing Alexandrov spaces while preserving the inherent curvature conditions. Similarly, Reshetnyak gluing methods efficiently tackle space amalgamations by extending the curvature bounds over composite structures.

Theoretical Implications and Dimensional Considerations

Alexandrov geometry provides a rich theoretical framework applicable to numerous mathematical domains, offering insights into spaces without traditional smoothness assumptions found in classical Riemannian geometry. This includes implications for dimension theory. The paper outlines varied dimensional analysis, including Hausdorff, topological, and linear dimensions, illustrating their interactions and hierarchical structures within Alexandrov spaces.

The investigation into $\CAT(0)$ and $\Alex(0)$ spaces is particularly noteworthy, as these spaces under zero curvature bounds serve as central structures, linking Alexandrov spaces to broader contexts, such as metric geometry and geometric group theory. The paper explores spaces formed under $\delta$-hyperbolic conditions and extends to the paper of group actions and their geometrical symmetry implications.

Future Directions

The text sets the stage for future explorations by suggesting extensions to the intrinsic theories delineated. Practical applications are largely set aside in this volume, pointing to a potential subsequent examination of Alexandrov spaces in applicable contexts such as general relativity, complex geometric analysis, or even practical computational geometry.

In conclusion, "Alexandrov Geometry: Foundations" provides a foundational yet nuanced approach to understanding spaces governed by curvature inequalities, revealing a wide array of geometric phenomena and providing tools for further mathematical exploration and application. It situates Alexandrov spaces as vital structures in contemporary geometry, promoting a deeper comprehension of space, curvature, and dimensional interplay.