Theorem of Bonnet-Schoenberg-Myers

The theorem of Bonnet-Schoenberg-Myers is a well-known theorem of the Géométrie riemannienne. It shows how local constraints on metric a riemannienne impose total conditions on the geometry of the variety. Its demonstration rests on a traditional use of the Formule of the second variation.

Theorem : If a riemannienne variety supplements has a sectionnelle curve undervalued by a strictly positive constant $\ delta$, then its diameter is limited by $\ pi \ sqrt \left\{\ delta\right\}$:
$\ forall X, K \left(X\right) \ geq \ delta>0 \ Rightarrow diam M \ Leq \ frac \left\{\ pi\right\} \left\{\ sqrt \left\{\ delta\right\}\right\}$
In particular, $M$ is compact.

The case of equality was studied:

Under the preceding notations, if the diameter of $M$ is equal to $\ pi \ sqrt \left\{K\right\}$, then $\left(M, G\right)$ is isometric with the Euclidean Sphère of ray $1 \ sqrt \left\{K\right\}$.

The theorem of Meyer has the following well-known corollary:

the fundamental group of riemannienne a compact variety of positive curve is a finished group.

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