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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