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Berger-Kazdan Comparison Theorem

The Berger-Kazdan comparison theorem gives a lower bound for the volume of a compact manifold, with equality only for a standard round sphere. Let M be a compact n-dimensional manifold with injectivity radius inj(M). Then

Vol(M)>=(c_ninj(M))/pi,

with equality iff M is isometric to the standard round sphere S^n with radius inj(M), where c_n(r) is the volume of the standard n-hypersphere of radius r.

See also

Blaschke Conjecture, Hypersphere, Injection, Isometry

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References

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

Referenced on Wolfram|Alpha

Berger-Kazdan Comparison Theorem

Cite this as:

Weisstein, Eric W. "Berger-Kazdan Comparison Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Berger-KazdanComparisonTheorem.html

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