During the last decades, several investigations were concerned with rigidity statements
for manifolds without conjugate points (some results can be found in the references). Based
on an idea by E. Hopf, K. Burns and G. Knieper proved that cylinders without conjugate
points and with a lower sectional curvature bound must be flat if the length of the shortest loop at
every point is globally bounded.
¶ The present article reduces the last condition to a limit for the asymptotic growth of loop-length
as the basepoint approaches the ends of the cylinder (Thm. 18). Along the way, the shape of cylinders
without conjugate points is characterized: The loop-length must be strictly monotone increasing to
both ends outside a – possibly empty – tube consisting of closed geodesics (Thm. 10).