▁is ▁the ▁general ▁linear ▁group ▁( or ▁the ▁orthogonal ▁group ▁in ▁classical ▁E uclidean ▁or ▁Riemann ian ▁geometry ). ▁Cart an ' s ▁ability ▁to ▁handle ▁many ▁other ▁types ▁of ▁fib ers ▁and ▁groups ▁allows ▁one ▁to ▁credit ▁him ▁with ▁the ▁first ▁general ▁idea ▁of ▁a ▁fi ber ▁bundle , ▁although ▁he ▁never ▁defined ▁it ▁explicitly . ▁This ▁concept ▁has ▁become ▁one ▁of ▁the ▁most ▁important ▁in ▁all ▁fields ▁of ▁modern ▁mathematics , ▁chief ly ▁in ▁global ▁differential ▁geometry ▁and ▁in ▁algebraic ▁and ▁differential ▁topology . ▁Cart an ▁used ▁it ▁to ▁form ulate ▁his ▁definition ▁of ▁a ▁connection , ▁which ▁is ▁now ▁used ▁univers ally ▁and ▁has ▁su pers eded ▁previous ▁attempts ▁by ▁several ▁geom eters , ▁made ▁after ▁ 1 9 1 7 , ▁to ▁find ▁a ▁type ▁of ▁" geometry " ▁more ▁general ▁than ▁the ▁Riemann ian ▁model ▁and ▁perhaps ▁better ▁adapted ▁to ▁a ▁description ▁of ▁the ▁universe ▁along ▁the ▁lines ▁of ▁general ▁relativ ity . ▁ ▁Cart an ▁showed ▁how ▁to ▁use ▁his ▁concept ▁of ▁connection ▁to ▁obtain ▁a ▁much ▁more ▁elegant ▁and ▁simple ▁presentation ▁of ▁Riemann ian ▁geometry . ▁His ▁chief ▁contribution ▁to ▁the ▁latter , ▁however , ▁was ▁the ▁discovery ▁and ▁study ▁of ▁the ▁symmetric ▁Riemann ▁spaces , ▁one ▁of ▁the ▁few ▁instances ▁in ▁which ▁the ▁initi ator ▁of ▁a ▁mathematical ▁theory ▁was ▁also ▁the ▁one ▁who ▁brought ▁it ▁to ▁its ▁completion . ▁Sym metric ▁Riemann ▁spaces ▁may ▁be ▁defined ▁in ▁various ▁ways , ▁the ▁simplest ▁of ▁which ▁post ul ates ▁the ▁existence ▁around ▁each ▁point ▁of ▁the