differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

, systolic freedom refers to the fact that closed

Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...

s may have arbitrarily small

volume Volume is a measure of occupied 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). The de ...

regardless of their

systolic Systole ( ) is the part of the cardiac cycle during which some chambers of the heart contract after refilling with blood. The term originates, via New Latin, from Ancient Greek (''sustolē''), from (''sustéllein'' 'to contract'; from ''sun ...

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, in Chișinău)Curriculum vitae
...

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 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 quantum computing that ...

. survey the main results on systolic freedom.

Example

The

complex projective plane In mathematics, the complex projective plane, usually denoted P2(C), 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 \mathbf^3,\qquad (Z_1, ...

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 In the mathematical field of Riemannian geometry, M. Gromov's systolic inequality bounds the length of the shortest non-contractible loop on a Riemannian manifold in terms of the volume of the manifold. Gromov's systolic inequality was proved in ...

References

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