In
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 ...
, Loewner's torus inequality is an
inequality due to
Charles Loewner
Charles Loewner (29 May 1893 – 8 January 1968) was an American mathematician. His name was Karel Löwner in Czech and Karl Löwner in German.
Early life and career
Karl Loewner was born into a Jewish family in Lany, about 30 km from Prag ...
. It relates the
systole
Systole ( ) is the part of the cardiac cycle during which some chambers of the heart contract after refilling with blood. Its contrasting phase is diastole, the relaxed phase of the cardiac cycle when the chambers of the heart are refilling ...
and the
area
Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
of an arbitrary
Riemannian metric
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 ...
on the
2-torus.
Statement
In 1949
Charles Loewner
Charles Loewner (29 May 1893 – 8 January 1968) was an American mathematician. His name was Karel Löwner in Czech and Karl Löwner in German.
Early life and career
Karl Loewner was born into a Jewish family in Lany, about 30 km from Prag ...
proved that every metric on the 2-
torus
In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
satisfies the optimal inequality
:
where "sys" is its
systole
Systole ( ) is the part of the cardiac cycle during which some chambers of the heart contract after refilling with blood. Its contrasting phase is diastole, the relaxed phase of the cardiac cycle when the chambers of the heart are refilling ...
, i.e. least length of a noncontractible loop. The constant appearing on the right hand side is the
Hermite constant
In mathematics, the Hermite constant, named after Charles Hermite, determines how long a shortest element of a lattice in Euclidean space can be.
The constant ''γn'' for integers ''n'' > 0 is defined as follows. For a lattice ''L'' in Euclidea ...
in dimension 2, so that Loewner's torus inequality can be rewritten as
:
The inequality was first mentioned in the literature in .
Case of equality
The boundary case of equality is attained if and only if the metric is flat and homothetic to the so-called ''equilateral torus'', i.e. torus whose group of deck transformations is precisely the
hexagonal lattice
The hexagonal lattice (sometimes called triangular lattice) is one of the five two-dimensional Bravais lattice types. The symmetry category of the lattice is wallpaper group p6m. The primitive translation vectors of the hexagonal lattice form an ...
spanned by the cube roots of unity in
. Geometrically, this torus can be obtained by gluing opposite pairs of edges of either a
regular hexagon
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A regular hexagon is de ...
, or a
rhombus
In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhom ...
with 60° and 120° angles.
Alternative formulation
Given a doubly periodic metric on
(e.g. an imbedding in
which is invariant by a
isometric action), there is a nonzero element
and a point
such that
, where
is a fundamental domain for the action, while
is the Riemannian distance, namely least length of a path joining
and
.
Proof of Loewner's torus inequality
Loewner's torus inequality can be proved most easily by using the computational formula for the variance,
:
Namely, the formula is applied to the
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...
defined by the measure of the unit area flat torus in the conformal class of the given torus. For the random variable ''X'', one takes the conformal factor of the given metric with respect to the flat one. Then the expected value E(''X''
2) of ''X''
2 expresses the total area of the given metric. Meanwhile, the expected value E(''X'') of ''X'' can be related to the systole by using
Fubini's theorem
In mathematical analysis, Fubini's theorem characterizes the conditions under which it is possible to compute a double integral by using an iterated integral. It was introduced by Guido Fubini in 1907. The theorem states that if a function is L ...
. The variance of ''X'' can then be thought of as the isosystolic defect, analogous to the isoperimetric defect of
Bonnesen's inequality. This approach therefore produces the following version of Loewner's torus inequality with isosystolic defect:
:
where ''ƒ'' is the conformal factor of the metric with respect to a unit area flat metric in its conformal class.
Higher genus
Whether or not the inequality
:
is satisfied by all surfaces of nonpositive
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's ...
is unknown. For
orientable surfaces of genus 2 and genus 20 and above, the answer is affirmative, see work by Katz and Sabourau below.
See also
*
Pu's inequality for the real projective plane
*
Gromov's systolic inequality for essential manifolds
*
Gromov's inequality for complex projective space
*
Eisenstein integer
In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are the complex numbers of the form
: z = a + b\omega ,
where and are integers and
: \omega = \frac ...
(an example of a hexagonal lattice)
*
Systoles of surfaces
References
*
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