▁in ▁ 1 8 5 5 ▁that ▁the ▁diameter ▁of ▁a ▁closed ▁surface ▁of ▁positive ▁Gaussian ▁curv ature ▁is ▁always ▁bounded ▁above ▁by ▁; ▁in ▁other ▁words ▁a ▁ge odes ic ▁real ising ▁the ▁metric ▁distance ▁between ▁two ▁points ▁cannot ▁have ▁length ▁greater ▁than ▁. ▁R ig id ity . ▁In ▁ 1 9 2 7 ▁C ohn - V ossen ▁proved ▁that ▁two ▁o val oid s ▁– ▁closed ▁surfaces ▁with ▁positive ▁Gaussian ▁curv ature ▁– ▁that ▁are ▁is ometric ▁are ▁necessarily ▁con gru ent ▁by ▁an ▁is ometry ▁of ▁. ▁Moreover , ▁a ▁closed ▁embedded ▁surface ▁with ▁positive ▁Gaussian ▁curv ature ▁and ▁constant ▁mean ▁curv ature ▁is ▁necessarily ▁a ▁sphere ; ▁like wise ▁a ▁closed ▁embedded ▁surface ▁of ▁constant ▁Gaussian ▁curv ature ▁must ▁be ▁a ▁sphere ▁( L ieb mann ▁ 1 8 9 9 ). ▁Hein z ▁H opf ▁showed ▁in ▁ 1 9 5 0 ▁that ▁a ▁closed ▁embedded ▁surface ▁with ▁constant ▁mean ▁curv ature ▁and ▁genus ▁ 0 , ▁i . e . ▁home omorphic ▁to ▁a ▁sphere , ▁is ▁necessarily ▁a ▁sphere ; ▁five ▁years ▁later ▁Alexand rov ▁removed ▁the ▁topological ▁assumption . ▁In ▁the ▁ 1 9 8 0 s , ▁W ente ▁constructed ▁imm ers ed ▁t ori ▁of ▁constant ▁mean ▁curv ature ▁in ▁E uclidean ▁ 3 - space . ▁Car ath é od ory ▁conject ure : ▁This ▁conject ure ▁states ▁that ▁a ▁closed ▁convex ▁three ▁times ▁differenti able ▁surface ▁adm its ▁at ▁least ▁two ▁um bil ic ▁points . ▁The ▁first ▁work