▁differential ▁forms ▁on ▁the ▁frame ▁bundle ▁and ▁formulas ▁involving ▁their ▁exterior ▁derivatives . ▁ ▁This ▁approach ▁is ▁particularly ▁simple ▁for ▁an ▁embedded ▁surface . ▁Thanks ▁to ▁a ▁result ▁of ▁, ▁the ▁connection ▁ 1 - form ▁on ▁a ▁surface ▁embedded ▁in ▁E uclidean ▁space ▁ ▁is ▁just ▁the ▁pull back ▁under ▁the ▁G auss ▁map ▁of ▁the ▁connection ▁ 1 - form ▁on ▁. ▁Using ▁the ▁identification ▁of ▁ ▁with ▁the ▁hom ogeneous ▁space ▁, ▁the ▁connection ▁ 1 - form ▁is ▁just ▁a ▁component ▁of ▁the ▁Ma urer – Cart an ▁ 1 - form ▁on ▁. ▁ ▁Global ▁differential ▁geometry ▁of ▁surfaces ▁Although ▁the ▁character isation ▁of ▁curv ature ▁involves ▁only ▁the ▁local ▁geometry ▁of ▁a ▁surface , ▁there ▁are ▁important ▁global ▁aspects ▁such ▁as ▁the ▁G auss – B on net ▁theorem , ▁the ▁uniform ization ▁theorem , ▁the ▁von ▁Mang old t - H ad am ard ▁theorem , ▁and ▁the ▁embed d ability ▁theorem . ▁There ▁are ▁other ▁important ▁aspects ▁of ▁the ▁global ▁geometry ▁of ▁surfaces . ▁These ▁include : ▁ ▁In ject ivity ▁radius , ▁defined ▁as ▁the ▁largest ▁ ▁such ▁that ▁two ▁points ▁at ▁a ▁distance ▁less ▁than ▁ ▁are ▁joined ▁by ▁a ▁unique ▁ge odes ic . ▁Wilhelm ▁K lingen berg ▁proved ▁in ▁ 1 9 5 9 ▁that ▁the ▁inject ivity ▁radius ▁of ▁a ▁closed ▁surface ▁is ▁bounded ▁below ▁by ▁the ▁minimum ▁of ▁ ▁and ▁the ▁length ▁of ▁its ▁smallest ▁closed ▁ge odes ic . ▁This ▁improved ▁a ▁theorem ▁of ▁Bon net ▁who ▁showed