Bishop–Gromov inequality explained
In mathematics, the Bishop–Gromov inequality is a comparison theorem in Riemannian geometry, named after Richard L. Bishop and Mikhail Gromov. It is closely related to Myers' theorem, and is the key point in the proof of Gromov's compactness theorem.[1]
Statement
Let
be a complete
n-dimensional Riemannian manifold whose
Ricci curvature satisfies the lower bound
for a constant
. Let
be the complete
n-dimensional
simply connected space of constant
sectional curvature
(and hence of constant Ricci curvature
); thus
is the
n-
sphere of radius
if
, or
n-dimensional
Euclidean space if
, or an appropriately rescaled version of
n-dimensional
hyperbolic space if
. Denote by
the ball of radius
r around a point
p, defined with respect to the Riemannian distance function.
Then, for any
and
, the function
is non-increasing on
.
As r goes to zero, the ratio approaches one, so together with the monotonicity this implies that
This is the version first proved by Bishop.
[2] [3] See also
Notes and References
- Book: Petersen, Peter. Riemannian Geometry. 3. Springer. 2016. Section 7.1.2. 978-3-319-26652-7.
- Bishop, R. A relation between volume, mean curvature, and diameter. Notices of the American Mathematical Society 10 (1963), p. 364.
- Bishop R.L., Crittenden R.J. Geometry of manifolds, Corollary 4, p. 256