Bishop-gromov Inequality
In
mathematics
, the
Bishop-Gromov inequality
is a classical theorem in
Riemannian geometry
. It is the key point in the proof of
Gromov's compactness theorem
.
Statement
Let us denote by
S^m_k
a
complete
simply connected
m
-
dimensional
Riemannian manifold
of constant
sectional curvature
k
, i.e. an
m
-
sphere
of radius
1/\sqrt{k}
if
k>0
,
Euclidean
m
-space
if
k=0
and
hyperbolic
m
-space
with curvature
k
if
k<0
. Let
M
be a complete
m
-dimensional Riemannian manifold with
Ricci curvature
\ge (m-1)k,
p\in M.
Let us denote by
v_p(R)
the volume of the ball with center
p
and radius
R
in
M
and by
V(R)
the volume of the ball of radius
R
in
S^m_k.
Then function
f_p(R)=v_p(R)/V(R)
is nonincreasing for any
p
. In particular this implies that for any
p
and
R
we have
v_p(R)\le V(R).
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