Let 
be the volume of a ball of radius 
in a complete 
-dimensional Riemannian
manifold with Ricci curvature tensor 
. Then 
, where 
is the volume of a ball in
a space having constant sectional curvature.
In addition, if equality holds for some ball, then this
ball is isometric to the ball of radius 
in the space of constant sectional
curvature 
.
Bishop's Inequality
See also
Ball, IsometryExplore with Wolfram|Alpha
References
Bishop, R. L. and Crittenden, R. Geometry of Manifolds. Providence, RI: Amer. Math. Soc., 2001.Chavel, I. Riemannian Geometry: A Modern Introduction. New York: Cambridge University Press, p. 123, 1994.Referenced on Wolfram|Alpha
Bishop's InequalityCite this as:
Weisstein, Eric W. "Bishop's Inequality." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BishopsInequality.html