Myers'S Theorem



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What is Myers's theorem?

The Myers theorem, also known as the Bonnet–Myers theorem, is a classical and well-known theorem in the mathematical field of Riemannian geometry. It was discovered by Sumner Byron Myers in 1941. Let (M,g) be a complete and smooth Riemannian manifold of dimension n. If k is a positive number with Ricg ≥ (n-1)k, then any curve of length greater than π/√k can be shortened. Precisely, this says that if γ : [a,b]→M is a smooth path of length greater than π/√k, then there exists ε > 0 and for each s ∈ (-ε,ε) a smooth path γs : [a,b]→M with γs(a)=γ(a) and with γs(b)=γ(b), with γ0 = γ, and such that the associated map (-ε,ε)×[a,b]→M is smooth, and such that the length of γs is less than that of γ for all s ∈ (0,ε). In a topological language, this says that there is a smooth homotopy of γ with fixed endpoints and which decreases length. An earlier result (from 1855), due to Ossian Bonnet, has the same conclusion but under the stronger assumption that the sectional curvatures is bounded below by k.

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