differential geometry w notes from teacher_Part_54

differential geometry w notes from teacher_Part_54 - •...

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Chapter 5 Lie Derivative 5.1 Lie Derivative of a Vector Field 5.1.1 Lie Bracket Let M be a manifold. Let X be a vector field on M . Let ϕ t : M M be the flow generated by X . Let x M . Then ϕ t ( x ) is the point on the integral curve of the vector field X going through x and such that ϕ 0 ( x ) = x and d ϕ t ( x ) dt = X ϕ t ( x ) . Let ϕ t * : T x M T ϕ t ( x ) M be the di erential of the di eomorphism ϕ t . Notice that ϕ 0 * = Id is the identity and ϕ - t * = ( ϕ t * ) - 1 is the inverse transformation. 107
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108 CHAPTER 5. LIE DERIVATIVE In local coordinates for small t we have ϕ i t ( x ) = x i + tX i ( x ) + O ( t 2 ) , so that ( ϕ t * ) i j = ∂ϕ i t ( x ) x j = δ i j + t X i ( x ) x j + O ( t 2 ) Thus ± d dt ϕ t * ! i j ² ² ² ² ² ² t = 0 = X i ( x ) x j . Let f be a smooth function on M . The flow ϕ t naturally defines a new function ( ϕ t * f )( x ) = ( f ϕ t )( x ) = f ( ϕ t ( x )) . Then for small t f ϕ = f + t X ( f ) + O ( t 2 )
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Unformatted text preview: . • Thus d dt ( f ◦ ϕ t ) ² ² ² ² t = = X ( f ) . • Let Y be another vector field on M . • Then d dt Y (( f ◦ ϕ t )) ² ² ² ² t = = Y ( X ( f )) . • Then at the point ϕ t ( x ) we have two di ff erent well defined vectors, Y ϕ t ( x ) and ϕ t * Y x . • Diagram. • Definition 5.1.1 A vector field Y is invariant under the flow ϕ t gener-ated by a vector field X if Y ϕ t ( x ) = ϕ t * Y x . An invariant vector field Y is also called a Jacobi field . di ff geom.tex; April 12, 2006; 17:59; p. 108...
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