[Paper Review] An invitation to Alexandrov geometry: CAT(0) spaces

Stephanie Alexander, Vitali Kapovitch+1 more|arXiv (Cornell University)|Jan 12, 2017

Geometric and Algebraic Topology10 references15 citations

TL;DR

This paper introduces CAT(0) spaces as a central concept in Alexandrov geometry, establishing that the Gromov–Hausdorff limit of a sequence of CAT[κ] spaces remains a CAT[κ] space under mild convergence conditions. The key contribution is a foundational stability result ensuring geometric properties are preserved in the limit, crucial for analyzing metric spaces with non-positive curvature.

ABSTRACT

Our goal is to show the beauty and power of Alexandrov geometry by reaching interesting applications and theorems with a minimum of preparation. The topics include 1. Reshetnyak's gluing theorem, 2. Estimates on the number of collisions in billiards, 3. Reshetnyak's majorization theorem, 4. Hadamard--Cartan globalization theorem, 5. Polyhedral spaces, 6. Construction of exotic aspherical manifolds, 7. The geometry of two-convex sets in Euclidean space, 8. Barycenters and dimension theory.

Motivation & Objective

• To provide an accessible introduction to Alexandrov geometry centered on CAT(0) spaces for researchers unfamiliar with the field.
• To establish the stability of CAT[κ] spaces under Gromov–Hausdorff convergence, a critical property for geometric analysis.
• To clarify that convergence conditions are sufficient as long as quadruples of points in the limit space can be approximated by those in the sequence of spaces.
• To lay the groundwork for further study of non-positively curved metric spaces through foundational topological and metric stability results.

Proposed method

• Utilizes the Gromov–Hausdorff convergence framework to analyze limits of metric spaces.
• Applies the definition of CAT[κ] spaces, which generalize non-positively curved Riemannian manifolds, to sequences of such spaces.
• Relies on the condition that any finite set of points in the limit space can be approximated arbitrarily well by corresponding point quadruples in the sequence of spaces.
• Employs the invariance of curvature bounds under limits, leveraging the intrinsic metric structure of CAT[κ] spaces.
• Uses the cone construction (U = Cone V) as a supporting example to illustrate geometric behavior in limit spaces.

Experimental results

Research questions

• RQ1 Under what conditions is the Gromov–Hausdorff limit of a sequence of CAT[κ] spaces itself a CAT[κ] space?
• RQ2 Does the choice of convergence definition affect the preservation of CAT[κ] properties in the limit?
• RQ3 Can arbitrary finite point configurations in the limit space be approximated by those in the approximating sequence of spaces?
• RQ4 How does the cone construction relate to the stability of CAT[κ] properties in geometric limits?

Key findings

• The Gromov–Hausdorff limit of a sequence of CAT[κ] spaces is always a CAT[κ] space, regardless of the specific convergence definition used, provided point quadruples in the limit can be approximated by those in the sequence.
• The stability of the CAT[κ] property under limits holds as long as the convergence ensures dense approximation of point configurations in the limit space.
• The result is robust to different convergence frameworks for metric spaces, as long as the approximation condition on quadruples is satisfied.
• The cone construction (U = Cone V) exemplifies a geometric object where such limit behavior can be analyzed, though its full implications are not detailed in the excerpt.