# Kazdan inequality

From Encyclopedia of Mathematics

Let be a real -dimensional scalar inner product space (cf. also Inner product; Pre-Hilbert space), let be the space of linear operators of , and let be a given family of symmetric linear operators depending continuously on . For , denote by the solution of the initial value problem

Suppose that is invertible for all . Then for every positive -function on satisfying on , one has

with equality if and only if for all .

#### References

[a1] | J.L. Kazdan, "An inequality arising in geometry" A.L. Besse (ed.) , Manifolds all of whose Geodesics are Closed , Springer (1978) pp. 243–246; Appendix E |

[a2] | I. Chavel, "Riemannian geometry: A modern introduction" , Cambridge Univ. Press (1995) |

**How to Cite This Entry:**

Kazdan inequality.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Kazdan_inequality&oldid=17407

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