▁infinity ▁towards ▁a ▁point ▁, ▁a ▁closed ▁disc ▁ ▁cent red ▁on ▁a ▁nearby ▁point ▁ ▁with ▁ ▁removed ▁would ▁be ▁contract ible ▁to ▁ ▁along ▁ge odes ics , ▁a ▁topological ▁impos s ibility . ▁ ▁Every ▁two ▁points ▁in ▁a ▁hom ot opy ▁class ▁are ▁connected ▁by ▁a ▁unique ▁ge odes ic ▁( see ▁above ). ▁ ▁Riemann ian ▁connection ▁and ▁parallel ▁transport ▁ ▁The ▁classical ▁approach ▁of ▁G auss ▁to ▁the ▁differential ▁geometry ▁of ▁surfaces ▁was ▁the ▁standard ▁elementary ▁approach ▁which ▁pred ated ▁the ▁emer gence ▁of ▁the ▁concepts ▁of ▁Riemann ian ▁manifold ▁initi ated ▁by ▁Bern hard ▁Riemann ▁in ▁the ▁mid - n in ete enth ▁century ▁and ▁of ▁connection ▁developed ▁by ▁Tul lio ▁Le vi - C iv ita , ▁É lie ▁Cart an ▁and ▁Hermann ▁W ey l ▁in ▁the ▁early ▁tw enti eth ▁century . ▁The ▁notion ▁of ▁connection , ▁cov ari ant ▁derivative ▁and ▁parallel ▁transport ▁gave ▁a ▁more ▁concept ual ▁and ▁uniform ▁way ▁of ▁understanding ▁curv ature , ▁which ▁not ▁only ▁allowed ▁general is ations ▁to ▁higher ▁dimensional ▁manif olds ▁but ▁also ▁provided ▁an ▁important ▁tool ▁for ▁defining ▁new ▁geometric ▁invari ants , ▁called ▁characteristic ▁classes . ▁The ▁approach ▁using ▁cov ari ant ▁derivatives ▁and ▁connections ▁is ▁now ad ays ▁the ▁one ▁adopted ▁in ▁more ▁advanced ▁text books . ▁ ▁Cov ari ant ▁derivative ▁Con ne ctions ▁on ▁a ▁surface ▁can ▁be ▁defined ▁from ▁various ▁equivalent ▁but ▁equally ▁important ▁points ▁of ▁view . ▁The ▁Riemann ian ▁connection ▁or ▁Le vi - C iv ita