# Berger inequality

From Encyclopedia of Mathematics

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 .

#### References

[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: http://encyclopediaofmath.org/index.php?title=Berger_inequality&oldid=15922

This article was adapted from an original article by H. Kaul (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article