An Alexandrov space is a metric space that satisfies a synthetic curvature bound in the sense of triangle comparison, originally introduced by Aleksandr Danilovich Alexandrov. The most common formulation requires the space to be a complete length space (i.e., any two points can be joined by a curve whose length equals the distance between the points) and to have sectional curvature bounded from below (or, dually, from above) by a real constant $ \kappa $.
Definition (curvature bounded below)
Let $(X,d)$ be a complete length space and let $\kappa\in\mathbb R$. For any geodesic triangle $\triangle xyz\subset X$ with side lengths $a=d(y,z),;b=d(z,x),;c=d(x,y)$, construct a comparison triangle $\tilde\triangle \tilde x\tilde y\tilde z$ in the simply‑connected, complete 2‑dimensional model space $M^2_\kappa$ of constant curvature $\kappa$ (the sphere $S^2_\kappa$ if $\kappa>0$, the Euclidean plane $\mathbb R^2$ if $\kappa=0$, or the hyperbolic plane $H^2_\kappa$ if $\kappa<0$) with the same side lengths $a,b,c$. The space $X$ is said to have curvature $\ge \kappa$ if for every point $p$ on a side of the triangle and the corresponding point $\tilde p$ on the same side of the comparison triangle, the distance to the opposite vertex satisfies
$$ d(p,\text{opposite vertex});\ge; d_{M^2_\kappa}(\tilde p,\text{opposite vertex}). $$
An analogous inequality with the opposite direction defines curvature $\le \kappa$. A space that fulfills the lower‑bound condition for some $\kappa$ is called an Alexandrov space with curvature bounded below by $\kappa$; similarly for an upper bound.
Main properties
| Property | Description |
|---|---|
| Geodesic | Every pair of points can be joined by a minimizing geodesic (a length‑realizing curve). |
| Local compactness | In the presence of a two‑sided curvature bound, Alexandrov spaces are locally compact and proper. |
| Tangent cones | At each point $x\in X$ there exists a unique metric cone $T_xX$ called the tangent cone, obtained as a Gromov–Hausdorff limit of rescaled neighborhoods of $x$. |
| Dimension | Alexandrov spaces admit an integer Hausdorff dimension; the regular part (points where the tangent cone is Euclidean) is open and dense. |
| Stability under limits | Gromov–Hausdorff limits of sequences of Alexandrov spaces with a uniform lower curvature bound are again Alexandrov spaces with the same bound (Gromov’s compactness theorem). |
| Splitting theorem | If an Alexandrov space with curvature $\ge 0$ contains a line (a bi‑infinite minimizing geodesic), it splits isometrically as a product $\mathbb R\times Y$. |
| Topological regularity | In dimensions $\le 4$ Alexandrov spaces are topological manifolds; in higher dimensions they may contain singular sets of codimension at least 2. |
Examples
- Riemannian manifolds whose sectional curvature satisfies $\sec\ge\kappa$ (or $\le\kappa$) are classical examples.
- Convex surfaces in Euclidean space. The intrinsic metric on a convex body’s boundary has curvature bounded below by 0.
- Metric quotients of manifolds by isometric group actions (orbifolds).
- Gromov–Hausdorff limits of sequences of Riemannian manifolds with a uniform lower curvature bound, such as Ricci‑limit spaces arising in Cheeger‑Colding theory.
- Polyhedral spaces equipped with the induced length metric on the 2‑skeleton of a Euclidean simplicial complex satisfy curvature $\ge 0$ in the sense of Alexandrov.
Historical context
The notion originated in the 1940s–1950s in Alexandrov’s work on the intrinsic geometry of surfaces and later on general metric spaces. His comparison‑triangle method provided a way to speak about curvature without requiring a smooth structure, laying the foundation for what is now called “synthetic geometry.” Systematic development of the theory was pursued in the 1990s by Burago, Gromov, Perelman, Petrunin, and many others, leading to a rich body of results that intersect Riemannian geometry, geometric group theory, and analysis on metric spaces.
Significance
Alexandrov spaces serve as a central class of objects in metric geometry and have become indispensable in several research areas:
- Geometric analysis – extension of concepts such as Laplacians, Sobolev spaces, and heat flow to nonsmooth settings.
- Ricci curvature bounds – via the Lott–Sturm–Villani theory of curvature‑dimension conditions, many results for Alexandrov spaces parallel those for manifolds with Ricci curvature bounds.
- Topology of manifolds – the stability and splitting theorems give powerful tools for understanding the topology of manifolds under collapsing or degeneration.
- Group actions – the structure of quotient spaces by isometries can be described within Alexandrov geometry, aiding the study of discrete groups acting on spaces of non‑positive curvature.
References (selected)
- A. D. Alexandrov, Intrinsic Geometry of Convex Surfaces, 1948.
- D. Burago, Y. Korevaar, and A. K. Lytchak, “Alexandrov Spaces with Curvature Bounded Below,” Geom. Funct. Anal. (1999).
- M. Gromov, Metric Structures for Riemannian and Non‑Riemannian Spaces, 1999.
- A. Petrunin, “Semiconcave Functions in Alexandrov Geometry,” Surveys in Differential Geometry (1998).
- J. Cheeger, T. H. Colding, “Lower Bounds on Ricci Curvature and the Almost Rigidity of Manifolds,” Ann. of Math. (1997).
These works, among many others, provide comprehensive treatments of the definition, properties, and applications of Alexandrov spaces.
