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 ...
, Pu's inequality, proved by
Pao Ming Pu, relates 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 surface homeomorphic to the
real projective plane
In mathematics, the real projective plane, denoted or , is a two-dimensional projective space, similar to the familiar Euclidean plane in many respects but without the concepts of distance, circles, angle measure, or parallelism. It is the sett ...
with the
lengths of the closed curves contained in it.
Statement
A student of
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 ...
, Pu proved in his 1950 thesis that every Riemannian surface
homeomorphic to the
real projective plane
In mathematics, the real projective plane, denoted or , is a two-dimensional projective space, similar to the familiar Euclidean plane in many respects but without the concepts of distance, circles, angle measure, or parallelism. It is the sett ...
satisfies the inequality
:
where
is 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 ...
of
.
The equality is attained precisely when the metric has constant
Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a smooth Surface (topology), surface in three-dimensional space at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
For ...
.
In other words, if all
noncontractible loops in
have length at least
, then
and the equality holds if and only if
is obtained from a Euclidean sphere of radius
by identifying each point with its antipodal.
Pu's paper also stated for the first time
Loewner's inequality, a similar result for Riemannian metrics on the
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 ...
.
Proof
Pu's original proof relies on the
uniformization theorem
In mathematics, the uniformization theorem states that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generali ...
and employs an averaging argument, as follows.
By uniformization, the Riemannian surface
is
conformally diffeomorphic to a round projective plane. This means that we may assume that the surface
is obtained from the Euclidean unit sphere
by identifying antipodal points, and the Riemannian length element at each point
is
:
where
is the Euclidean length element and the function
, called the conformal factor, satisfies
.
More precisely, the universal cover of
is
, a loop
is noncontractible if and only if its lift
goes from one point to its opposite, and the length of each curve
is
:
Subject to the restriction that each of these lengths is at least
, we want to find an
that minimizes the
:
where
is the upper half of the sphere.
A key observation is that if we average several different
that satisfy the length restriction and have the same area
, then we obtain a better conformal factor
, that also satisfies the length restriction and has
:
:
and the inequality is strict unless the functions
are equal.
A way to improve any non-constant
is to obtain the different functions
from
using
rotations of the sphere
, defining
. If we
average over all possible rotations, then we get an
that is constant over all the sphere. We can further reduce this constant to minimum value
allowed by the length restriction. Then we obtain the obtain the unique metric that attains the minimum area
.
Reformulation
Alternatively, every metric on the sphere
invariant under the antipodal map admits a pair of opposite points
at Riemannian distance
satisfying
A more detailed explanation of this viewpoint may be found at the page
Introduction to systolic geometry.
Filling area conjecture
An alternative formulation of Pu's inequality is the following. Of all possible fillings of the
Riemannian circle
In mathematics, a metric circle is the metric space of arc length on a circle, or equivalently on any rectifiable simple closed curve of bounded length. The metric spaces that can be embedded into metric circles can be characterized by a four-p ...
of length
by a
-dimensional disk with the strongly isometric property, the round
hemisphere
Hemisphere may refer to:
In geometry
* Hemisphere (geometry), a half of a sphere
As half of Earth or any spherical astronomical object
* A hemisphere of Earth
** Northern Hemisphere
** Southern Hemisphere
** Eastern Hemisphere
** Western Hemi ...
has the least area.
To explain this formulation, we start with the observation that the equatorial circle of the unit
-sphere
is a
Riemannian circle
In mathematics, a metric circle is the metric space of arc length on a circle, or equivalently on any rectifiable simple closed curve of bounded length. The metric spaces that can be embedded into metric circles can be characterized by a four-p ...
of length
. More precisely, the Riemannian distance function
of
is induced from the ambient Riemannian distance on the sphere. Note that this property is not satisfied by the standard imbedding of the unit circle in the Euclidean plane. Indeed, the Euclidean distance between a pair of opposite points of the circle is
only
, whereas in the Riemannian circle it is
.
We consider all fillings of
by a
-dimensional disk, such that the metric induced by the inclusion of the circle as the boundary of the disk is the Riemannian
metric of a circle of length
. The inclusion of the circle as the boundary is then called a strongly isometric imbedding of the circle.
Gromov
conjectured
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), have sh ...
that the round hemisphere gives the "best" way of filling the circle even when the filling surface is allowed to have positive genus .
Isoperimetric inequality
Pu's inequality bears a curious resemblance to the classical
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'' ...
:
for
Jordan curves in the plane, where
is the length of the curve while
is the area of the region it bounds. Namely, in both cases a 2-dimensional quantity (area) is bounded by (the square of) a 1-dimensional quantity (length). However, the inequality goes in the opposite direction. Thus, Pu's inequality can be thought of as an
"opposite" isoperimetric inequality.
See also
*
Filling area conjecture In differential geometry, Mikhail Gromov's filling area conjecture asserts that the hemisphere has minimum area among the orientable surfaces that fill a closed curve of given length without introducing shortcuts between its points.
Definitions ...
*
Gromov's systolic inequality for essential manifolds
*
Gromov's inequality for complex projective space
*
Loewner's torus 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 ...
*
Systoles of surfaces
References
*
*
*
*
*
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Riemannian geometry
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