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 {\ delta}

\ forall X, K (X) \ geq \ delta>0 \ Rightarrow diam M \ Leq \ frac {\ pi} {\ sqrt {\ delta}}

In particular,

M

is compact.

The case of equality was studied:

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

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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