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]

Statement[edit]

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 Mkn be the n-dimensional simply connected Riemannian space form of constant sectional curvature k =  ρ/(n-1) (i.e. constant Ricci curvature ρ), so Mkn 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(pr) the ball of radius r around a point p, defined with respect to the Riemannian distance function.

Then for any p ∈ M and pk ∈ Mkn the function

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

is non-increasing on (0, ∞).

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

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

This is the original Bishop's inequality[2][3]

See also[edit]

References[edit]

  1. ^ Petersen, Peter (2006). "Chapter 9". Riemannian Geometry (2 ed.). Springer. ISBN 978-0-387-29246-5. 
  2. ^ Bishop, R. A relation between volume, mean curvature, and diameter. Amer. Math. Soc. Not. 10 (1963), p. 364.
  3. ^ Corollary 4, p. 256 in Bishop R.L., Crittenden R.J. Geometry of manifolds