ard ' s ▁theorem ▁that ▁a ▁simply ▁connected ▁surface ▁of ▁non - pos itive ▁curv ature ▁is ▁home omorphic ▁to ▁the ▁plane , ▁are ▁equally ▁valid ▁in ▁this ▁more ▁general ▁setting . ▁ ▁Alexand rov ' s ▁comparison ▁inequality ▁ ▁The ▁simplest ▁form ▁of ▁the ▁comparison ▁inequality , ▁first ▁proved ▁for ▁surfaces ▁by ▁Alexand rov ▁around ▁ 1 9 4 0 , ▁states ▁that ▁ ▁The ▁inequality ▁follows ▁from ▁the ▁fact ▁that ▁if ▁ ▁describes ▁a ▁ge odes ic ▁paramet riz ed ▁by ▁ar cl ength ▁and ▁ ▁is ▁a ▁fixed ▁point , ▁then ▁ ▁is ▁a ▁convex ▁function , ▁i . e . ▁ ▁T aking ▁ge odes ic ▁polar ▁coordinates ▁with ▁origin ▁at ▁ ▁so ▁that ▁, ▁convex ity ▁is ▁equivalent ▁to ▁ ▁Ch anging ▁to ▁normal ▁coordinates ▁, ▁ ▁at ▁, ▁this ▁inequality ▁becomes ▁, ▁where ▁ ▁corresponds ▁to ▁the ▁unit ▁vector ▁. ▁This ▁follows ▁from ▁the ▁inequality ▁, ▁a ▁consequence ▁of ▁the ▁non - neg ativity ▁of ▁the ▁derivative ▁of ▁the ▁W rons k ian ▁of ▁ ▁and ▁ ▁from ▁St urm – Li ou ville ▁theory . ▁ ▁Ex istence ▁of ▁ge odes ics ▁On ▁a ▁complete ▁cur ved ▁surface ▁any ▁two ▁points ▁can ▁be ▁joined ▁by ▁a ▁ge odes ic . ▁This ▁is ▁a ▁special ▁case ▁of ▁the ▁H opf – R in ow ▁theorem , ▁which ▁also ▁applies ▁in ▁higher ▁dimensions . ▁The ▁comple ten ess ▁assumption ▁is ▁automatically ▁ful filled ▁for ▁a ▁surface ▁which ▁is ▁embedded ▁as ▁a ▁closed ▁subset ▁of ▁E uclidean ▁space .