Systolic geometry

Systolic geometry studies how the lengths of shortest noncontractible curves relate to volume and other geometric properties of spaces.

Riemannian metric: A smoothly varying inner product on tangent spaces that determines lengths, angles, and volumes on a manifold. Systolic lengths and volumes are measured using a Riemannian metric.
Systole: The length of a shortest noncontractible closed curve in a metric space, with variants defined by different homotopy or homology classes. It is the basic length quantity in systolic inequalities.
Loewner's torus inequality: The inequality stating that every Riemannian metric on a two-torus satisfies area at least (2/√3) times the squared homotopy systole. It is the sharp basic example of a systolic area bound.
Charles Loewner: A Czech-American mathematician whose work included the sharp systolic inequality for the two-torus. His torus inequality became a foundational result in the subject.
Essential manifold: A closed manifold whose fundamental class remains nonzero under the map to the classifying space of its fundamental group. Essentiality is the topological condition behind major volume-systole bounds.
Closed curve: A continuous path whose initial and terminal points coincide. Systoles are lengths of selected closed curves.
Homotopy systole: The length of the shortest closed curve that is not contractible. This is the fundamental-group version of the systole.
Pu's inequality: The inequality stating that every Riemannian metric on the real projective plane has area at least (2/π) times the squared homotopy systole. It gives a sharp area bound for a nonorientable surface.
Pólya's conjecture: The conjecture that the first Laplace eigenvalue times the area of a flat torus is at most 2π². Its torus optimization problem helped shape early questions about geometric extremality.
Homological systolic inequality: An inequality bounding a manifold's volume using lengths or masses of cycles representing nontrivial homology classes. It generalizes curve-based bounds to higher-dimensional homology.

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