inextensible flows of timelike curves with sabban frame in s2

10 V. Asil, T. Körpinar, S. Ba³Then, a straightforward computation using above system gives( )∂α ∂fS= 1∂t ∂σ + fS 2 t + ψs,( )∂s ∂fS= 3∂t ∂σ + fS 2 κ g t − ψα,⟨ ⟩ ∂αwhere ψ =∂t , s . Thus, we obtain the assertion of the theorem.The following theorem states the conditions on the curvature and torsionfor the ow to be inextensible.Theorem 3.5. Let ∂α be inextensible. Then the system of partial dierential∂tequationsholds.∂κ g∂σ + ψ = ∂2 f S 3∂σ 2 + ∂ ∂s (fS 2 κ g ),( ∂ 2 f S )1κ g ψ = −∂σ 2 + ∂fS 2∂σProof. Assume that ∂α is inextensible. Then∂t∂ ∂t= ∂ [( ) ( ) ]∂fS1∂fS∂σ ∂t ∂σ ∂σ + fS 2 α + 3∂σ + fS 2 κ g s( ∂ 2 f S ) ( ) ( )1=∂σ 2 + ∂fS 2 ∂fSα + [ 1∂fS∂σ ∂σ + fS 2 +κ 3 g∂σ + fS 2 κ g ]t( ∂ 2 f S 3+∂σ 2 + ∂ )∂σ (fS 2 κ g ) s.From the Sabban frame, we have∂ ∂t∂t ∂σ = ∂ ∂t (α+κ gs)=[ ∂κg∂σ + ψ ]s +[ ( ) ( ) ]∂fS1∂fS∂σ + fS 2 + κ 3 g∂σ + fS 2 κ g t − κ g ψα.Therefore,∂κ g∂σ + ψ = ∂2 f S 3∂σ 2 + ∂∂σ (fS 2 κ g )