Lee claims in his book that $S^1\times \Bbb R$ (considered as a submanifold of $\Bbb R^3$) admits no conjugate points along any geodesic.
I am struggling to make that rigorous. Being conjugate along a geodesic $c$ means that there is a nontrivial Jacobi field $J$ which vanishes at the end points. Each such Jacobi field is induced by a variation of $c$ of the form $$c_s(t)=\exp_p(t(c'(0)+sW))$$ for some nonzero $W\in T_pM$. How can I see that such a variation must have $W=0$?
Take a geodesic in the cylinder and a Jacobi field on it, and pull these back to the universal cover - this is $\Bbb R^2$, and geodesics are straight lines. Because $\Bbb R^2$ is flat, we see that the Jacobi condition is just that $J''(t) = 0$ (taking derivatives with respect to $t$), and hence that $J(t) = v+wt$. This could only vanish at both endpoints of the geodesic if it was identically zero.