In differential geometry, systolic freedom refers to the fact that closed Riemannian manifold

Riemannian manifold

In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

s may have arbitrarily small volume

Volume

Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

 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.E.S. preprint in 1992 (which eventually appeared as ), and was further developed by Mikhail Katz

Mikhail Katz

Mikhail Katz is an Israeli mathematician, a professor of mathematics at Bar Ilan University. His main interests are differential geometry and geometric topology; he is the author of a book about systolic geometry....

, Michael Freedman

Michael Freedman

Michael Hartley Freedman is a 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 Poincaré conjecture. Freedman and Robion Kirby showed that an exotic R4 manifold exists.Freedman was born...

 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

Quantum error correction is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is essential if one is to achieve fault-tolerant quantum computation that can deal not only with noise on stored quantum...

. survey the main results on systolic freedom.

Example

The complex projective plane 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

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...

.