Berger inequality

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For a compact Riemannian manifold , let

where is the ball around with radius , be the injectivity radius, and set . Then the inequality

holds, with equality if and only if is isometric to the standard sphere with diameter .

This inequality relies on the Kazdan inequality applied to the Jacobi equation for operators on for a unit vector . Here, is the curvature operator, is the parallel transport along the geodesic ray , and is the parallel translated curvature operator on .


[a1] M. Berger, "Une borne inférieure pour le volume d'une variété riemannienes en fonction du rayon d'injectivité" Ann. Inst. Fourier (Grenoble) , 30 (1980) pp. 259–265
[a2] I. Chavel, "Riemannian geometry: A modern introduction" , Cambridge Univ. Press (1995)
How to Cite This Entry:
Berger inequality. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by H. Kaul (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article