▁the ▁most ▁varied ▁problems ▁in ▁differential ▁geometry , ▁Lie ▁groups , ▁analyt ical ▁dynamics , ▁and ▁general ▁relativ ity . ▁He ▁discussed ▁a ▁large ▁number ▁of ▁examples , ▁tre ating ▁them ▁in ▁an ▁extremely ▁elli ptic ▁style ▁that ▁was ▁made ▁possible ▁only ▁by ▁his ▁un c anny ▁algebraic ▁and ▁geometric ▁insight . ▁ ▁D iffer ential ▁geometry ▁Cart an ' s ▁contributions ▁to ▁differential ▁geometry ▁are ▁no ▁less ▁impress ive , ▁and ▁it ▁may ▁be ▁said ▁that ▁he ▁rev ital ized ▁the ▁whole ▁subject , ▁for ▁the ▁initial ▁work ▁of ▁Riemann ▁and ▁Dar b oux ▁was ▁being ▁lost ▁in ▁dre ary ▁comput ations ▁and ▁minor ▁results , ▁much ▁as ▁had ▁happened ▁to ▁elementary ▁geometry ▁and ▁invariant ▁theory ▁a ▁generation ▁earlier . ▁His ▁gu iding ▁principle ▁was ▁a ▁considerable ▁extension ▁of ▁the ▁method ▁of ▁" mov ing ▁frames " ▁of ▁Dar b oux ▁and ▁Rib auc our , ▁to ▁which ▁he ▁gave ▁a ▁trem end ous ▁flex ibility ▁and ▁power , ▁far ▁beyond ▁anything ▁that ▁had ▁been ▁done ▁in ▁classical ▁differential ▁geometry . ▁In ▁modern ▁terms , ▁the ▁method ▁consists ▁in ▁associ ating ▁to ▁a ▁fi ber ▁bundle ▁E ▁the ▁principal ▁fi ber ▁bundle ▁having ▁the ▁same ▁base ▁and ▁having ▁at ▁each ▁point ▁of ▁the ▁base ▁a ▁fi ber ▁equal ▁to ▁the ▁group ▁that ▁acts ▁on ▁the ▁fi ber ▁of ▁E ▁at ▁the ▁same ▁point . ▁If ▁E ▁is ▁the ▁tang ent ▁bundle ▁over ▁the ▁base ▁( which ▁since ▁Lie ▁was ▁essentially ▁known ▁as ▁the ▁manifold ▁of ▁" contact ▁elements "), ▁the ▁corresponding ▁group