differential geometry w notes from teacher_Part_58

# differential geometry w notes from teacher_Part_58 - L X...

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5.2. LIE DERIVATIVE OF FORMS AND TENSORS 115 1. Computation in local coordinates. ± The Lie derivative of a 1-form ω with respect to a vector ﬁeld X can be deﬁned in an intrinsic way. L X ω is a 1-form whose value on any vector ﬁeld Y is ( L X ω )( Y ) = X ( ω ( Y )) - ω ([ X , Y ]) . More generally, we have Theorem 5.2.4 Let X and Y 1 , . . . , Y p be vector ﬁelds on a manifold M and α Λ p be a p-form. Then L X ± α ( Y 1 , . . . , Y p ) ² = ( L X α )( Y 1 , . . . , Y p ) + p X i = 1 α ( Y 1 , . . . , L X Y i , . . . , Y p ) . Proof : By induction or direct calculation. ± Remark. This can be used as an intrinsic deﬁnition of L X α . Theorem 5.2.5 Let X and Y be any two vector ﬁelds and c R . Then 1. L X + Y = L X + L Y , 2. L c X = cL X , 3. L X Y = - L Y X , 4. [ L X , L Y ] = L [ X , Y ] . Proof : 1. ± di geom.tex; April 12, 2006; 17:59; p. 115

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116 CHAPTER 5. LIE DERIVATIVE Theorem 5.2.6 Let g i j be a Riemannian metric on an n-dimensional manifold M and vol = p | g | dx 1 ∧ ··· ∧ dx n be the Riemannian volume form. Let X be a vector ﬁeld on M. Then L X vol = ( div X )vol , where div X is a scalar function deﬁned by div X = *
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Unformatted text preview: L X vol = 1 p | g | i p | g | X i . Proof : 1. Direct calculation. Use L X g i j . The scalar div X is called the divergence of the vector eld X . Remark. Let Y 1 , . . . , Y n be vector elds invariant under the vector eld X . Then div X = ( L X vol )( Y 1 , . . . , Y n ) vol ( Y 1 , . . . , Y n ) = d dt log vol ( Y 1 , . . . , Y n ) t = . Thus, div X is the logarithmic rate of change of the volume along the ow. Remark. Let n-1 be an ( n-1)-form dened by = i X vol . In components i 1 ... i n-1 = X j p | g | ji 1 ... i n-1 Then d = ( div X )vol . To prove this compute in local coordinates ( d ) i 1 ... i n = n [ i 1 X j p | g | ji 2 ... i n di geom.tex; April 12, 2006; 17:59; p. 116...
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## differential geometry w notes from teacher_Part_58 - L X...

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