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 (n-1) K \,

for a constant K\in \mathbb{R}. Let M_K^n be the complete n-dimensional simply connected space of constant sectional curvature K (and hence of constant Ricci curvature (n-1)K); thus M_K^n is the n-sphere of radius 1/\sqrt{K} if K > 0, or n-dimensional Euclidean space if K=0, or an appropriately rescaled version of 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\in M and p_K\in M_K^n, 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 version first proved by Bishop,[2][3] originally assuming the (unnecessary) added hypothesis that r is less than the injectivity radius at p.

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. ^ Bishop R.L., Crittenden R.J. Geometry of manifolds, Corollary 4, p. 256