We will discuss various situations in which the Dirac operator on spin manifolds can be used to infer distance estimates from lower scalar curvature bounds. This includes Gromov’s question on the distance between the boundary components of „bands“ $M \times [-1,1]$, rigidity properties of warped products, and distance estimates in the context of Witten’s spinor proof of the positive mass theorem. We will contrast this with an obstruction in the non-spin setting concerning the non-existence of complete positive scalar curvature metrics on certain manifolds with at least two ends.

Based on joint works with S. Cecchini and D. Räde.