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Mathhhttidgsslfw , from this point we cared dohelhegeodesrqnhb , Ckrauk relation for surfaces fmvottm , ek , bit first he page to give an dtnahechrdrizhmf geodesics as Straightest paths . To do so , we need to define covariant differentiation & parallel transport . Df : v : z→|R3 B a smooth vector field on I if Hp ) ETPE t PEE & ✓ ( Ilu , D) B a smooth film f the lad cards u&v . We can differentiate V along E , but intrinsic Y we af see the pork f the dentate target to E . Def . The covariant denwthe direct ml downed DWV = project n of DWEHP onto Tp E = DWV - < Dwu , i ) ri Similarly , we an diftmthte veotrtdds defied only alga are 8 in -2 , ht only in te 8 ' direction : py , ,+ , VHHH =fk VHHH pointed to THE = at HHHI - L # until , nisn We say a vector field V day 8 , } prdld if 8 , # V = 0 . Lenny : V B parallel day 8 ⇐ dat VHHH B in the indian . t# On the unit sphne , ht 8 be the greet circle in the xy - plane . 1hm both the tagutuedvfdd TH & the field 10,0 , 1) are parallel day 8 . Now , agidg an arbitrary vectr field Unit = aluirhutbhnlxv & ht B = lwu , wu ) . lets wok at Owvkn ) . dropped th & mi bk they 're mthendireth Diftrntstijm the u directing .ws y pay V = a Ilutaxuutbixutbxru - lnormlsttf ) = auiut al Tuiiu + hit . ) + bixrtb ( Puixnttuixr ) = ( autatuitb Rilxn + I but a hit b Rift, Similarly , % V = 1 art a hit b Ii ) # 1 but a hit b Pri ) * That , Ow V = Wu 1 QIV ) + Wu 1 QIU )
