differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

, systolic freedom refers to the fact that closed

Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

s may have arbitrarily small

volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...

regardless of their systolic invariants. That is, systolic invariants or products of systolic invariants do not in general provide universal (i.e. curvature-free) lower bounds for the total volume of a closed Riemannian manifold. Systolic freedom was first detected by Mikhail Gromov in an I.H.É.S. preprint in 1992 (which eventually appeared as ), and was further developed by

Mikhail Katz Mikhail "Mischa" Gershevich Katz (, ; born 1958)Curriculum vitae
retrieved ...

Michael Freedman Michael Hartley Freedman (born April 21, 1951) is an American mathematician at Microsoft Station Q, a research group at the University of California, Santa Barbara. In 1986, he was awarded a Fields Medal for his work on the 4-dimensional gen ...

and others. Gromov's observation was elaborated on by . One of the first publications to study systolic freedom in detail is by . Systolic freedom has applications in

quantum error correction Quantum error correction (QEC) is a set of techniques used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is theorised as essential to achieve fault tolerant ...

. survey the main results on systolic freedom.

Example

The

complex projective plane In mathematics, the complex projective plane, usually denoted or is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates :(Z_1,Z_2,Z_3) \in \C^3, \qquad (Z_1,Z_2, ...

admits Riemannian metrics of arbitrarily small volume, such that every essential surface is of area at least 1. Here a surface is called "essential" if it cannot be contracted to a point in the ambient 4-manifold.

Systolic constraint

The opposite of systolic freedom is systolic constraint, characterized by the presence of systolic inequalities such as Gromov's systolic inequality for essential manifolds.

References

*. ''Astérisque'' 216, Exp. No. 771, 5, 279–310. *. *. *. *. *. *. {{Systolic geometry navbox Differential geometry Riemannian geometry Systolic geometry