Bishop–Gromov inequality

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.

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

for a constant ${\displaystyle K\in \mathbb {R} }$. Let ${\displaystyle M_{K}^{n}}$ be the complete n-dimensional simply connected space of constant sectional curvature ${\displaystyle K}$ (and hence of constant Ricci curvature ${\displaystyle (n-1)K}$); thus ${\displaystyle M_{K}^{n}}$ is the n-sphere of radius ${\displaystyle 1/{\sqrt {K}}}$ if K > 0, or n-dimensional Euclidean space if ${\displaystyle K=0}$, or an appropriately rescaled version of n-dimensional hyperbolic space if ${\displaystyle K<0}$. Denote by B(p, r) the ball of radius r around a point p, defined with respect to the Riemannian distance function.

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