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 Myers's theorem, also known as the Bonnet–Myers theorem, is a celebrated, fundamental theorem in the mathematical field of Riemannian geometry. It was discovered by Sumner Byron Myers in 1941. It asserts the following: In the special case of ...
, and is the key point in the proof of Gromov's compactness theorem.


Statement

Let M be a complete ''n''-dimensional Riemannian manifold whose
Ricci curvature In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...
satisfies the lower bound : \mathrm \geq (n-1) K for a constant K\in \R. Let M_K^n be the complete ''n''-dimensional simply connected space of constant
sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a poi ...
K (and hence of constant Ricci curvature (n-1)K); thus M_K^n is the ''n''-
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
of radius 1/\sqrt if K>0, or ''n''-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
if K=0, or an appropriately rescaled version of ''n''-dimensional
hyperbolic space In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. The ...
if K<0. 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 p\in M and p_K\in M_K^n, the function : \phi(r) = \frac is non-increasing on (0,\infty). As ''r'' goes to zero, the ratio approaches one, so together with the monotonicity this implies that : \mathrm \,B(p,r) \leq \mathrm \, B(p_K,r). This is the version first proved by Bishop.Bishop R.L., Crittenden R.J. Geometry of manifolds, Corollary 4, p. 256


See also

*
Comparison theorem In mathematics, comparison theorems are theorems whose statement involves comparisons between various mathematical objects of the same type, and often occur in fields such as calculus, differential equations and Riemannian geometry. Differential eq ...
* Gromov's inequality (disambiguation)


References

{{DEFAULTSORT:Bishop-Gromov Inequality Riemannian geometry Geometric inequalities