▁, ▁because ▁the ▁right ▁hand ▁side ▁has ▁integral ▁ 0 ▁by ▁the ▁G auss – B on net ▁theorem . ▁Thus ▁ ▁satisfies ▁ ▁away ▁from ▁. ▁It ▁follows ▁that ▁ ▁is ▁a ▁complete ▁metric ▁of ▁constant ▁curv ature ▁ 0 ▁on ▁the ▁complement ▁of ▁, ▁which ▁is ▁therefore ▁is ometric ▁to ▁the ▁plane . ▁Compos ing ▁with ▁s tere ographic ▁projection , ▁it ▁follows ▁that ▁there ▁is ▁a ▁smooth ▁function ▁ ▁such ▁that ▁ ▁has ▁Gaussian ▁curv ature ▁+ 1 ▁on ▁the ▁complement ▁of ▁. ▁The ▁function ▁ ▁automatically ▁extends ▁to ▁a ▁smooth ▁function ▁on ▁the ▁whole ▁of ▁. ▁ ▁Sur faces ▁of ▁non - pos itive ▁curv ature ▁In ▁a ▁region ▁where ▁the ▁curv ature ▁of ▁the ▁surface ▁satisfies ▁, ▁ge odes ic ▁tri angles ▁satisfy ▁the ▁C AT ( 0 ) ▁in equal ities ▁of ▁comparison ▁geometry , ▁studied ▁by ▁Cart an , ▁Alexand rov ▁and ▁Top on og ov , ▁and ▁considered ▁later ▁from ▁a ▁different ▁point ▁of ▁view ▁by ▁Bru hat ▁and ▁T its ; ▁thanks ▁to ▁the ▁vision ▁of ▁G rom ov , ▁this ▁character isation ▁of ▁non - pos itive ▁curv ature ▁in ▁terms ▁of ▁the ▁underlying ▁metric ▁space ▁has ▁had ▁a ▁prof ound ▁impact ▁on ▁modern ▁geometry ▁and ▁in ▁particular ▁geometric ▁group ▁theory . ▁Many ▁results ▁known ▁for ▁smooth ▁surfaces ▁and ▁their ▁ge odes ics , ▁such ▁as ▁B irk hoff ' s ▁method ▁of ▁construct ing ▁ge odes ics ▁by ▁his ▁curve - short ening ▁process ▁or ▁van ▁Mang old t ▁and ▁Had am