▁connection ▁is ▁perhaps ▁most ▁easily ▁understood ▁in ▁terms ▁of ▁lif ting ▁vector ▁fields , ▁considered ▁as ▁first ▁order ▁differential ▁operators ▁acting ▁on ▁functions ▁on ▁the ▁manifold , ▁to ▁differential ▁operators ▁on ▁the ▁tang ent ▁bundle ▁or ▁frame ▁bundle . ▁In ▁the ▁case ▁of ▁an ▁embedded ▁surface , ▁the ▁lift ▁to ▁an ▁operator ▁on ▁vector ▁fields , ▁called ▁the ▁cov ari ant ▁derivative , ▁is ▁very ▁simply ▁described ▁in ▁terms ▁of ▁orthogonal ▁projection . ▁Indeed , ▁a ▁vector ▁field ▁on ▁a ▁surface ▁embedded ▁in ▁ ▁can ▁be ▁regarded ▁as ▁a ▁function ▁from ▁the ▁surface ▁into ▁. ▁Another ▁vector ▁field ▁acts ▁as ▁a ▁differential ▁operator ▁component - wise . ▁The ▁resulting ▁vector ▁field ▁will ▁not ▁be ▁tang ent ▁to ▁the ▁surface , ▁but ▁this ▁can ▁be ▁corrected ▁taking ▁its ▁orthogonal ▁projection ▁onto ▁the ▁tang ent ▁space ▁at ▁each ▁point ▁of ▁the ▁surface . ▁As ▁Ric ci ▁and ▁Le vi - C iv ita ▁real ised ▁at ▁the ▁turn ▁of ▁the ▁tw enti eth ▁century , ▁this ▁process ▁depends ▁only ▁on ▁the ▁metric ▁and ▁can ▁be ▁locally ▁expressed ▁in ▁terms ▁of ▁the ▁Christ off el ▁symbols . ▁ ▁Par allel ▁transport ▁Par allel ▁transport ▁of ▁tang ent ▁vectors ▁along ▁a ▁curve ▁in ▁the ▁surface ▁was ▁the ▁next ▁major ▁advance ▁in ▁the ▁subject , ▁due ▁to ▁Le vi - C iv ita . ▁It ▁is ▁related ▁to ▁the ▁earlier ▁notion ▁of ▁cov ari ant ▁derivative , ▁because ▁it ▁is ▁the ▁mon od rom y ▁of ▁the ▁ordinary ▁differential ▁equation ▁on ▁the ▁curve ▁defined ▁by ▁the ▁cov ari