1.(10 points) Let (R 2/ , d) be the unit cylinder. Let A = (0, 0) and B = (, z) be two points on the cylinder, where z 6= 0. Use Proposition 2.1.15 to (1) show that there are infinitely many geodesics connecting A and B; (2) find the lengths of these geodesics

Recall that locally Euclidean spaces can be constructed from an equivalence relation in R2. In these cases, we have seen that the quotient map 1) : R2 > R2/ ~ gives an isometry from BAA) to B1,. (p(A)} for r > 0 small and any A E R2. Proposition 2.1.15. Lei (139/ ~43) is as above. Let 1: be a straight line in 13?. Then (1)1307 is a. geodesic in 1R2/ N. (2) length(qv|;) = length((p 0 TN!) for any closed and bounded interval I C R. Let (R2 / ~, d) be the unit cylinder. Let A = (0,0) and B = (0, z) be two points on the cylinder, where z * 0. Use Proposition 2.1.15 to (1) show that there are infinitely many geodesics connecting A and B; (2) find the lengths of these geodesics