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Bishop–Gromov inequality

Statement[edit]

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

${\displaystyle \mathrm {Ric} \geq (n-1)K}$

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${\displaystyle K>0}$, or n-dimensional Euclidean spaceif${\displaystyle K=0}$, or an appropriately rescaled version of n-dimensional hyperbolic spaceif${\displaystyle K<0}$. Denote by ${\displaystyle B(p,r)}$ the ball of radius r around a point p, defined with respect to the Riemannian distance function.

Then, for any ${\displaystyle p\in M}$ and ${\displaystyle p_{K}\in M_{K}^{n}}$, the function

${\displaystyle \phi ($r)={\frac {\mathrm {Vol} \,B(p,r)}{\mathrm {Vol} \,B(p_{K},r)}}}

is non-increasing on ${\displaystyle (0,\infty )}$.

Asr goes to zero, the ratio approaches one, so together with the monotonicity this implies that

${\displaystyle \mathrm {Vol} \,B(p,r)\leq \mathrm {Vol} \,B(p_{K},r).}$

This is the version first proved by Bishop.^[2][3]

See also[edit]

• Comparison theorem
• Gromov's inequality (disambiguation)

References[edit]

1. ^ Petersen, Peter (2016). "Section 7.1.2". Riemannian Geometry (3 ed.). Springer. ISBN 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.