# 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 …