Bishop–Gromov inequality

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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]^[2]

[edit] Statement

Let $M$ be a complete n-dimensional Riemannian manifold whose Ricci curvature satisfies the lower bound

$\mathrm{Ric} \geq \rho \,$

for a constant ρ ∈ R. Further, let M_kⁿ be the n-dimensional simply connected Riemannian space form of constant sectional curvature k = (n-1) ρ (i.e. constant Ricci curvature ρ), so M_kⁿ is an n-sphere if k > 0, it is an n-dimensional Euclidean space if k = 0, and it is an n-dimensional hyperbolic space if k < 0. Denote by B(p, r) the ball of radius r around a point p , defined with respect to the Riemannian distance function.

Then for any p ∈ M and p_k ∈ M_kⁿ the function

$\phi(r) = \frac{\mathrm{Vol} \, B(p,r)}{\mathrm{Vol}\, B(p_k,r)}$

is non-increasing on (0, ∞). Furthermore, as r goes to zero, the ratio approaches one, so together with the monotonicty this implies that

$\mathrm{Vol} \,B(p,r) \le \mathrm{Vol} \, B(p_k,r).$

^ Cheeger, J.; Ebin, D.G. (2008). Comparison theorems in Riemannian geometry. Providence, RI: AMS Chelsea Publishing. ISBN 978-0-8218-4417-5. MR 2394158.
^ Petersen, Peter (2006). "Chapter 9". Riemannian Geometry (2 ed.). Springer. ISBN 978-0-387-29246-5.