In
mathematics, Berger's isoembolic inequality is a result in
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...
that gives a lower bound on the volume of a
Riemannian manifold and also gives a
necessary and sufficient condition
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
for the manifold to be
isometric to the -
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
al
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
with its usual "round" metric. The theorem is named after the
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
Marcel Berger
Marcel Berger (14 April 1927 – 15 October 2016) was a French mathematician, doyen of French differential geometry, and a former director of the Institut des Hautes Études Scientifiques (IHÉS), France. Formerly residing in Le Castera in Las ...
, who derived it from an inequality proved by
Jerry Kazdan
Jerry Lawrence Kazdan (born 31 October 1937 in Detroit, Michigan) is an American mathematician noted for his work in differential geometry and the study of partial differential equations. His contributions include the Berger–Kazdan comparis ...
.
Statement of the theorem
Let be a
closed -dimensional Riemannian manifold with
injectivity radius
This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology.
The following articles may also be useful; they either contain specialised vocabulary or provi ...
. Let denote the Riemannian volume of and let denote the volume of the standard -dimensional sphere of radius one. Then
:
with equality
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
is isometric to the
-sphere with its usual round metric. This result is known as Berger's ''isoembolic inequality''. The proof relies upon an analytic inequality proved by Kazdan. The original work of Berger and Kazdan appears in the appendices of
Arthur Besse's book "Manifolds all of whose geodesics are closed." At this stage, the isoembolic inequality appeared with a non-optimal constant. Sometimes Kazdan's inequality is called ''Berger–Kazdan inequality''.
References
Books.
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External links
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Geometric inequalities
Theorems in Riemannian geometry
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