Riemannian manifolds in the large

Aspects of global Riemannian geometry

“…However, he seems not to have pointed out that for complete manifolds this gives diameter bounds. Later in [131], after Myers in [98] published his generalization of Bonnet's diameter bound, Synge then points out that this is indeed a trivial conclusion and also berates Schoenberg for his paper [125], where Schoenberg obtains conjugate point estimates similar to those in Synge's 1925 paper. What is even more interesting is that Synge correctly points out that it is in fact like shooting flies with cannon balls to get these diameter bounds from conjugate point estimates, for one can simply show directly from the second variation formula that geodesics that are too long can't minimize length.…”

Obstruction to prescribed positive Ricci curvature

“…Moreover, the regularity argument above, now applied to h, combined with a bootstrap argument, provides a harmonic atlas in which {X, h) is a C°° Riemannian manifold. So Myers' theorem [10] holds for (X, h), contradicting the noncompactness of X. D Proof of Theorem 3.…”

Hilbert Manifolds Without Epiconjugate Points

“…This theorem had been proved by Myers [6] under the assumption that M was analytic, and this assumption was essential in his proof. Kobayashi, and also Helgason in his book [3], showed that analyticity was superfluous and could be replaced with mere smoothness of a sufficiently high order.…”