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,from this point we cared dohelhegeodesrqnhb , Ckrauk

relation for surfaces fmvottm,ek

, bit first hepage

togive an dtnahechrdrizhmf geodesics as Straightestpaths .

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 tPEE & ✓ ( Ilu ,D) B a smooth film fthe lad cards u&v .

We can differentiate V along E,but intrinsicY we af see the pork f the dentate target to E

.

Def.The covariant denwthe directmldowned

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 inte 8

'

direction :

py,,+,VHHH =fk VHHH pointed to THE = at HHHI - L # until

,nisn

Wesay 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 boththe 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 ).

droppedth & mi bk they're mthendireth

Diftrntstijmthe 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 )