Strengthened injectivity radius bounds for manifolds with positive scalar curvature

Green’s inequality shows that a compact Riemannian manifold with scalar curvature at least $n(n-1)$ has injectivity radius at most $\pi$, and that equality is achieved only for the radius 1 sphere. In this work we show how extra topological assumptions can lead to stronger upper bounds. The topologies we consider are ${\mathbb{S}}^2\times {\mathbb{T}}^{n-k-2}\times {\mathbb{R}}^k$ for $n\le 7$ and $0\le k\le 2$ and 3-manifolds with positive scalar curvature except lens spaces $L(p,q)$ with $p$ odd. We also prove a strengthened inequality for $3$-manifolds with positive scalar curvature and large diameter. Our proof uses previous results of Gromov and Zhu.