inextensible flows of timelike curves with sabban frame in s2
inextensible flows of timelike curves with sabban frame in s2 inextensible flows of timelike curves with sabban frame in s2
8 V. Asil, T. Körpinar, S. Ba³Lemma 3.2. Let ∂α∂t = fS 1 α + fS 2 t + fS 3 s be a smooth ow of the timelike curveα. The ow is inextensible if and only if∂v∂t + ∂fS 2∂u = −fS 1 v − f S 3 vκ g . (3.3)Proof. Suppose that ∂α be a smooth ow of the timelike curve α. Using∂tdenition of α, we have⟨ ∂αv 2 =∂u , ∂α ⟩. (3.4)∂uBy dierentiating of the formula (??), we get2v ∂v∂t = ∂ ⟨ ∂α∂t ∂u , ∂α ⟩.∂uOn the other hand, changing ∂∂u and ∂ , we have∂tv ∂v ⟨ ⟩∂α∂t = ∂u , ∂∂u (∂α ∂t ) .From (3.1), we obtainv ∂v ⟨ ∂α∂t = ∂u , ∂ (fS∂u 1 α + f S 2 t + f S 3 s )⟩ .By the Sabban formula, we have⟨ ( ) () ( ) ⟩∂v ∂fS∂t = t, 1∂u + fS 2 v α + f S 1 v + ∂fS 2∂fS∂u + fS 3 vκ g t + 3∂u + fS 2 vκ g s .Making necessary calculations, from above equation we obtain (3.3) whichproves the lemma.Theorem 3.3. Let ∂α∂t = fS 1 α + fS 2 t + fS 3 s be a smooth ow of the timelikecurve α. The ow is inextensible if and only if∂f S 2∂u = −fS 3 vκ g − f S 1 v.
Inextensible ows of timelike curves... 9Proof. Assume that ∂α∂t∂∂t σ(u, t) = u ∂v0∂t du = u 0is inextensible. From (3.2), we have(−f S 1 v − ∂fS 2∂u − fS 3 vκ g)du = 0. (3.5)Substituting (3.3) in (3.5) completes the proof of the theorem.Now we restrict ourselves to arc length parametrized curves. That is,v = 1 and the local coordinate u corresponds to the curve arc length σ. Werequire the following lemma.Lemma 3.4. The following relations hold( )∂t ∂fS= 1∂t ∂σ + fS 2 α +(∂α ∂fS= 1∂t ∂σ + fS 2where ψ =∂s∂t⟨ ⟩ ∂α∂t , s .=)t + ψs,( ∂fS3∂σ + fS 2 κ g)t − ψα,Proof. Using denition of α, we have∂t∂t = ( ∂fS1∂σ + fS 2( ∂fS3∂σ + fS 2 κ g)s,∂t∂t = ∂ ∂α∂t ∂σ = ∂∂σ (fS 1 α + f S 2 t + f S 3 s).Using the Sabban equations, we nd that)α +Substituting (3.3) in (3.6), we get∂t∂t = ( ∂fS1∂σ + fS 2(f S 1 + ∂fS 2∂σ + fS 3 κ g)t +)α +( ∂fS3∂σ + fS 2 κ g)s.Now dierentiate the Sabban frame by t:∂f S ⟨1∂σ + fS 2 + t, ∂α ⟩= 0,∂t∂f S ⟨3∂σ + fS 2 κ g + t, ∂s ⟩= 0,∂t⟨ψ + α, ∂s ⟩= 0.∂t( ∂fS3∂σ + fS 2 κ g)s. (3.6)
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8 V. Asil, T. Körp<strong>in</strong>ar, S. Ba³Lemma 3.2. Let ∂α∂t = fS 1 α + fS 2 t + fS 3 s be a smooth ow <strong>of</strong> the <strong>timelike</strong> curveα. The ow is <strong><strong>in</strong>extensible</strong> if and only if∂v∂t + ∂fS 2∂u = −fS 1 v − f S 3 vκ g . (3.3)Pro<strong>of</strong>. Suppose that ∂α be a smooth ow <strong>of</strong> the <strong>timelike</strong> curve α. Us<strong>in</strong>g∂tdenition <strong>of</strong> α, we have⟨ ∂αv 2 =∂u , ∂α ⟩. (3.4)∂uBy dierentiat<strong>in</strong>g <strong>of</strong> the formula (??), we get2v ∂v∂t = ∂ ⟨ ∂α∂t ∂u , ∂α ⟩.∂uOn the other hand, chang<strong>in</strong>g ∂∂u and ∂ , we have∂tv ∂v ⟨ ⟩∂α∂t = ∂u , ∂∂u (∂α ∂t ) .From (3.1), we obta<strong>in</strong>v ∂v ⟨ ∂α∂t = ∂u , ∂ (fS∂u 1 α + f S 2 t + f S 3 s )⟩ .By the Sabban formula, we have⟨ ( ) () ( ) ⟩∂v ∂fS∂t = t, 1∂u + fS 2 v α + f S 1 v + ∂fS 2∂fS∂u + fS 3 vκ g t + 3∂u + fS 2 vκ g s .Mak<strong>in</strong>g necessary calculations, from above equation we obta<strong>in</strong> (3.3) whichproves the lemma.Theorem 3.3. Let ∂α∂t = fS 1 α + fS 2 t + fS 3 s be a smooth ow <strong>of</strong> the <strong>timelike</strong>curve α. The ow is <strong><strong>in</strong>extensible</strong> if and only if∂f S 2∂u = −fS 3 vκ g − f S 1 v.
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