Definition
Systolic geometry is a branch of differential geometry and geometric topology that studies quantitative relationships between the volume (or more generally, the Riemannian metric) of a manifold and the length of its shortest non‑contractible closed geodesic, called the systole. The central objects of interest are inequalities that bound the volume from below in terms of powers of the systole, known as systolic inequalities.
Overview
The field originated from questions posed by Charles Loewner in the 1940s concerning the optimal lower bound for the area of a torus in terms of the length of its shortest non‑contractible loop. Loewner proved that for any Riemannian metric on the 2‑torus $T^{2}$, the inequality
$$
\operatorname{area}(T^{2}) \geq \frac{\sqrt{3}}{2},\operatorname{sys}(T^{2})^{2}
$$
holds, with equality attained for the flat equilateral torus. This result inaugurated the systematic study of systolic problems.
Subsequent work extended systolic considerations to higher‑dimensional manifolds, non‑orientable surfaces, and manifolds with additional structure (e.g., symplectic or complex). Notable contributions include:
- Gromov’s systolic inequality (1983), which for an essential $n$-dimensional closed Riemannian manifold $M$ asserts
$$ \operatorname{vol}(M) \geq C_{n},\operatorname{sys}(M)^{n}, $$
where $C_{n}>0$ depends only on the dimension. - Results on stable systoles, which replace the ordinary systole by a homology‑theoretic invariant obtained from the mass norm on real homology.
- Development of isosystolic concepts, investigating manifolds for which the systolic inequality becomes an equality up to a universal constant.
Research in systolic geometry intersects with minimal surface theory, filling invariants, metric geometry, and the geometry of groups.
Etymology/Origin
The term systole derives from the Greek word “συστός” (system) meaning “tight” or “compressed,” reflecting the notion of the shortest essential loop in a space. The adjective systolic thus denotes properties related to this minimal length. The discipline’s name reflects its focus on quantitative (“systolic”) aspects of geometric structures.
Characteristics
Key characteristics of systolic geometry include:
- Systolic Invariants – Primary invariants such as the (homotopy) systole $\operatorname{sys}_\pi(M)$ and the (homology) stable systole $\operatorname{stsys}_k(M)$ measure minimal lengths (or volumes) of non‑trivial cycles.
- Systolic Inequalities – Inequalities linking volume (or higher‑dimensional volume of cycles) to powers of systolic invariants, often with universal constants dependent on topology or dimension.
- Essential Manifolds – Manifolds whose fundamental class maps non‑trivially to the classifying space of its fundamental group; these are the primary subjects of Gromov’s inequality.
- Metric Dependence – Systolic quantities depend on the chosen Riemannian metric; the field studies extremal metrics that minimize volume for a fixed systole or maximize systole for a fixed volume.
- Connections to Filling Volume – The filling radius and filling volume concepts, introduced by Gromov, are closely related to systolic estimates.
Related Topics
- Riemannian Geometry – Underlying smooth metric structures.
- Geodesic Length Spectrum – Study of lengths of closed geodesics, of which the systole is the first element.
- Minimal Surface Theory – Techniques for constructing area‑minimizing representatives of homology classes.
- Metric Geometry – Includes concepts such as Gromov–Hausdorff convergence and curvature bounds relevant to systolic problems.
- Geometric Group Theory – The notion of essential manifolds ties into group homology and asphericity.
- Filling Invariants – Filling radius, filling volume, and related quantities introduced by Gromov.
- Isoperimetric Inequalities – Classical inequalities that compare volume and boundary area; systolic inequalities can be viewed as a non‑linear analogue.
Systolic geometry continues to be an active research area, offering deep connections between topology, analysis, and geometry.