In the
mathematical
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
field of
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...
,
M. Gromov's systolic inequality bounds the length of the shortest
non-contractible loop on a
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 ...
in terms of the volume of the manifold. Gromov's systolic inequality was proved in 1983;
[see ] it can be viewed as a generalisation, albeit non-optimal, of
Loewner's torus inequality and
Pu's inequality for the real projective plane.
Technically, let ''M'' be an
essential Riemannian manifold of dimension ''n''; denote by sys''π''
1(''M'') the homotopy 1-systole of ''M'', that is, the least length of a non-contractible loop on ''M''. Then Gromov's inequality takes the form
:
where ''C''
''n'' is a universal constant only depending on the dimension of ''M''.
Essential manifolds
A closed manifold is called ''essential'' if its
fundamental class defines a nonzero element in the
homology of its
fundamental group
In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...
, or more precisely in the homology of the corresponding
Eilenberg–MacLane space
In mathematics, specifically algebraic topology, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name. ...
. Here the fundamental class is taken in homology with integer coefficients if the manifold is orientable, and in coefficients modulo 2, otherwise.
Examples of essential manifolds include
aspherical manifolds,
real projective space
In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in the real space It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space.
Basic properti ...
s, and
lens spaces.
Proofs of Gromov's inequality
Gromov's original 1983 proof is about 35 pages long. It relies on a number of techniques and inequalities of global Riemannian geometry. The starting point of the proof is the imbedding of X into the Banach space of Borel functions on X, equipped with the sup norm. The imbedding is defined by mapping a point ''p'' of ''X'', to the real function on ''X'' given by the distance from the point ''p''. The proof utilizes the
coarea inequality, the
isoperimetric inequality
In mathematics, the isoperimetric inequality is a geometric inequality involving the square of the circumference of a closed curve in the plane and the area of a plane region it encloses, as well as its various generalizations. '' Isoperimetric'' ...
, the cone inequality, and the deformation theorem of
Herbert Federer
Herbert Federer (July 23, 1920 – April 21, 2010) was an American mathematician. He is one of the creators of geometric measure theory, at the meeting point of differential geometry and mathematical analysis.Parks, H. (2012''Remembering Herbert F ...
.
Filling invariants and recent work
One of the key ideas of the proof is the introduction of filling invariants, namely the
filling radius In Riemannian geometry, the filling radius of a Riemannian manifold ''X'' is a metric invariant of ''X''. It was originally introduced in 1983 by Mikhail Gromov (mathematician), Mikhail Gromov, who used it to prove his Gromov's systolic inequality ...
and the filling volume of ''X''. Namely, Gromov proved a sharp inequality relating the systole and the filling radius,
:
valid for all essential manifolds ''X''; as well as an inequality
:
valid for all closed manifolds ''X''.
It was shown by that the filling invariants, unlike the systolic invariants, are independent of the topology of the manifold in a suitable sense.
and developed approaches to the proof of Gromov's systolic inequality for essential manifolds.
Inequalities for surfaces and polyhedra
Stronger results are available for surfaces, where the asymptotics when the genus tends to infinity are by now well understood, see
systoles of surfaces. A uniform inequality for arbitrary 2-complexes with non-free fundamental groups is available, whose proof relies on the
Grushko decomposition theorem.
Notes
See also
*
Filling area conjecture
*
Gromov's inequality (disambiguation)
*
Gromov's inequality for complex projective space
*
Loewner's torus inequality
*
Pu's inequality
*
Systolic geometry
In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and ...
References
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{{Riemannian geometry
Geometric inequalities
Riemannian geometry
Systolic geometry
Theorems in Riemannian geometry