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.[1]

Statement

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

for a constant . Let be the complete n-dimensional simply connected space of constant sectional curvature (and hence of constant Ricci curvature ); thus is the n-sphere of radius if K > 0, or n-dimensional Euclidean space if , or an appropriately rescaled version of n-dimensional hyperbolic space if . 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 and , the function

is non-increasing on (0, ∞).

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

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

See also

References

  1. ↑ Petersen, Peter (2016). "Section 11.1.3". Riemannian Geometry (3 ed.). Springer. ISBN 978-3-319-26652-7.
  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
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