▁piece wise ▁ge odes ic ▁curves . ▁ ▁The ▁connection ▁can ▁thus ▁be ▁described ▁in ▁terms ▁of ▁lif ting ▁paths ▁in ▁the ▁manifold ▁to ▁paths ▁in ▁the ▁tang ent ▁or ▁orth on ormal ▁frame ▁bundle , ▁thus ▁form alis ing ▁the ▁classical ▁theory ▁of ▁the ▁" mov ing ▁frame ", ▁fav oured ▁by ▁French ▁authors . ▁L ifts ▁of ▁loops ▁about ▁a ▁point ▁give ▁rise ▁to ▁the ▁hol onomy ▁group ▁at ▁that ▁point . ▁The ▁Gaussian ▁curv ature ▁at ▁a ▁point ▁can ▁be ▁recovered ▁from ▁parallel ▁transport ▁around ▁increasing ly ▁small ▁loops ▁at ▁the ▁point . ▁Equ ival ently ▁curv ature ▁can ▁be ▁calculated ▁directly ▁at ▁an ▁inf init es imal ▁level ▁in ▁terms ▁of ▁Lie ▁brackets ▁of ▁lifted ▁vector ▁fields . ▁ ▁Connection ▁ 1 - form ▁The ▁approach ▁of ▁Cart an ▁and ▁W ey l , ▁using ▁connection ▁ 1 - forms ▁on ▁the ▁frame ▁bundle ▁of ▁, ▁gives ▁a ▁third ▁way ▁to ▁understand ▁the ▁Riemann ian ▁connection . ▁They ▁noticed ▁that ▁parallel ▁transport ▁dict ates ▁that ▁a ▁path ▁in ▁the ▁surface ▁be ▁lifted ▁to ▁a ▁path ▁in ▁the ▁frame ▁bundle ▁so ▁that ▁its ▁tang ent ▁vectors ▁lie ▁in ▁a ▁special ▁sub space ▁of ▁cod im ension ▁one ▁in ▁the ▁three - dimensional ▁tang ent ▁space ▁of ▁the ▁frame ▁bundle . ▁The ▁projection ▁onto ▁this ▁sub space ▁is ▁defined ▁by ▁a ▁differential ▁ 1 - form ▁on ▁the ▁orth on ormal ▁frame ▁bundle , ▁the ▁connection ▁form . ▁This ▁enabled ▁the ▁curv ature ▁properties ▁of ▁the ▁surface ▁to ▁be ▁encoded ▁in