Bishop–Gromov inequality

In mathematics, the Bishop–Gromov inequality is a classical theorem in Riemannian geometry, named after Richard L. Bishop and Mikhail Gromov. It is the key point in the proof of Gromov's compactness theorem.

tatement

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 .

Let $M$ be a complete "m"-dimensional Riemannian manifold with Ricci curvature $ge\; (m-1)k,$ $pin\; M.$

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

:$v\_p(R)le\; V(R).$

ee also

* Comparison theorem
* Gromov's inequality