Positive scalar curvature and the conjugate radius

A classical result in Riemannian geometry is the theorem of P. O. Bonnet and S. B. Myers stating that a complete Riemannian \(n\)-manifold \(M\) with Ricci curvature bounded from below by \(n-1\) has diameter at most \(\pi\).

In the introduction of Bo Zhu’s recent preprint arXiv:2201.12668 the following ‘analogue’ of the Bonnet-Myers Theorem for scalar curvature is recalled: If \(M\) is a closed Riemannian \(n\)-manifold with scalar curvature bounded from below by \(n(n-1)\), then the conjugate radius of \(M\) is at most \(\pi\).

Recall that the notion of the conjugate radius is derived from the notion of conjugate points. The latter is defined as follows: A point \(q\) is conjugate to a point \(p\) if it is a critical value of the exponential map based at \(p\).

Now although this result that Zhu recalls is from 1963 (proven by L. W. Green, https://doi.org/10.2307/1970344), I was not aware of it. Quite a shame, since recently it became en vogue to study quantitative implications of positive scalar curvature bounds (e.g., band width estimates) and here we have a prime example of such a result which is almost 60 years old already. Maybe I should study a bit more ‘classical’ Riemannian geometry …