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Bishop's Inequality

Let $V(r)$ be the volume of a Ball of radius $r$ in a complete $n$-D Riemannian Manifold with Ricci Curvature $\geq (n-1)\kappa$. Then $V(r)\geq V_\kappa(r)$, where $V_\kappa$ 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 $r$ in the space of constant Sectional Curvature $\kappa$.


Chavel, I. Riemannian Geometry: A Modern Introduction. New York: Cambridge University Press, 1994.

© 1996-9 Eric W. Weisstein