inextensible flows of timelike curves with sabban frame in s2

Inextensible ows of timelike curves... 11andThus, we obtain the theorem.( ∂ 2 f S )1κ g ψ = −∂σ 2 + ∂fS 2.∂σCorollary 3.6.( ) ∂fSκ 1 g∂σ + fS 2 = − ∂fS 3∂σ − fS 2 κ g − ∂ψ∂σ .Proof. Similarly, we have∂ ∂s∂σ ∂t= ∂ [( ) ]∂fS3∂σ ∂σ + fS 2 κ g t − ψα[ ( ∂ 2 f S 3=∂σ + ∂) ( )( )fS ∂fS∂σ 2 κ g − ψ t+κ 3 g∂σ + fS 2 κ g s[ ( )∂fS+ 3∂σ + fS 2 κ g − ∂ψ ] ]α .∂σOn the other hand, a straightforward computation gives∂ ∂s∂t ∂σ = ∂ ∂t (κ gt)= ∂κ g∂t t + κ g[( ) ( ) ]∂fS1∂fS∂σ + fS 2 α + 3∂σ + fS 2 κ g s .Combining these equalities, we obtain the corollary.In the light of Theorem 3.5, we express the following corollaries withoutproofs.Corollary 3.7.Corollary 3.8.∂κ g∂t = ∂2 f S 3∂σ + ∂ ( )fS∂σ 2 κ g − ψ.∂f S 3∂σ + fS 2 κ g − ∂ψ∂σ = ∂fS 1∂σ + fS 2 .References[1] G. Chirikjian, J. Burdick, A modal approach to hyper-redundant manipulatorkinematics, IEEE Trans. Robot. Autom., 10, 343354 (1994).