▁rank ▁ 2 . ▁These ▁closed ▁surfaces ▁have ▁no ▁is ometric ▁embed d ings ▁in ▁. ▁They ▁do ▁never theless ▁admit ▁is ometric ▁embed d ings ▁in ▁; ▁in ▁the ▁easiest ▁case ▁this ▁follows ▁from ▁the ▁fact ▁that ▁the ▁tor us ▁is ▁a ▁product ▁of ▁two ▁circles ▁and ▁each ▁circle ▁can ▁be ▁is omet r ically ▁embedded ▁in ▁. ▁ ▁S pher ical ▁geometry ▁ ▁The ▁is ometry ▁group ▁of ▁the ▁unit ▁sphere ▁ ▁in ▁ ▁is ▁the ▁orthogonal ▁group ▁, ▁with ▁the ▁rotation ▁group ▁ ▁as ▁the ▁subgroup ▁of ▁is omet ries ▁pres erving ▁orientation . ▁It ▁is ▁the ▁direct ▁product ▁of ▁ ▁with ▁the ▁ant ip od al ▁map , ▁sending ▁ ▁to ▁. ▁The ▁group ▁ ▁acts ▁trans it ively ▁on ▁. ▁The ▁stabil izer ▁subgroup ▁of ▁the ▁unit ▁vector ▁( 0 , 0 , 1 ) ▁can ▁be ▁identified ▁with ▁, ▁so ▁that ▁. ▁ ▁The ▁ge odes ics ▁between ▁two ▁points ▁on ▁the ▁sphere ▁are ▁the ▁great ▁circle ▁ar cs ▁with ▁these ▁given ▁end points . ▁If ▁the ▁points ▁are ▁not ▁ant ip od al , ▁there ▁is ▁a ▁unique ▁short est ▁ge odes ic ▁between ▁the ▁points . ▁The ▁ge odes ics ▁can ▁also ▁be ▁described ▁group ▁theoret ically : ▁each ▁ge odes ic ▁through ▁the ▁North ▁pole ▁( 0 , 0 , 1 ) ▁is ▁the ▁orbit ▁of ▁the ▁subgroup ▁of ▁rot ations ▁about ▁an ▁axis ▁through ▁ant ip od al ▁points ▁on ▁the ▁equ ator . ▁ ▁A ▁spher ical ▁triangle ▁is ▁a ▁ge odes