▁However , ▁it ▁is ▁no ▁longer ▁ful filled ▁if , ▁for ▁example , ▁we ▁remove ▁an ▁isolated ▁point ▁from ▁a ▁surface . ▁For ▁example , ▁the ▁complement ▁of ▁the ▁origin ▁in ▁the ▁E uclidean ▁plane ▁is ▁an ▁example ▁of ▁a ▁non - complete ▁surface ; ▁in ▁this ▁example ▁two ▁points ▁which ▁are ▁diam etr ically ▁opposite ▁across ▁the ▁origin ▁cannot ▁be ▁joined ▁by ▁a ▁ge odes ic ▁without ▁leaving ▁the ▁pun ct ured ▁plane ). ▁ ▁Von ▁Mang old t – H ad am ard ▁theorem ▁For ▁closed ▁surfaces ▁of ▁non - pos itive ▁curv ature , ▁von ▁Mang old t ▁( 1 8 8 1 ) ▁and ▁Had am ard ▁( 1 8 9 8 ) ▁proved ▁that ▁the ▁exponential ▁map ▁at ▁a ▁point ▁is ▁a ▁covering ▁map , ▁so ▁that ▁the ▁universal ▁covering ▁space ▁of ▁the ▁manifold ▁is ▁. ▁This ▁result ▁was ▁general ised ▁to ▁higher ▁dimensions ▁by ▁Cart an ▁and ▁is ▁usually ▁referred ▁to ▁in ▁this ▁form ▁as ▁the ▁Cart an – H ad am ard ▁theorem . ▁For ▁surfaces , ▁this ▁result ▁follows ▁from ▁three ▁important ▁facts : ▁ ▁The ▁exponential ▁map ▁has ▁non - zero ▁Jacob ian ▁everywhere ▁for ▁non - pos it ively ▁cur ved ▁surfaces , ▁a ▁consequence ▁of ▁the ▁non - van ishing ▁of ▁. ▁ ▁Every ▁ge odes ic ▁is ▁infinitely ▁extend ible , ▁a ▁result ▁known ▁as ▁the ▁H opf – R in ow ▁theorem ▁for ▁- dimensional ▁manif olds . ▁In ▁two ▁dimensions , ▁if ▁a ▁ge odes ic ▁t ended ▁at