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differential geometry w notes from teacher_Part_57

# differential geometry w notes from teacher_Part_57 - 113...

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5.2. LIE DERIVATIVE OF FORMS AND TENSORS 113 Proposition 5.2.2 The Lie derivative of a p-form α with respect to a vector field X is given by ( L X α ) i 1 ... i p = X j j α i 1 ... i p + α ji 2 ... i p i 1 X j + · · · + α i 1 ... i p - 1 j i p X j Proof : 1. Use that ( ϕ * t α ) i 1 ... i p ( x ) = ∂ϕ j 1 t ( x ) x i 1 · · · ∂ϕ j p t ( x ) x i p α j 1 ... j p ( ϕ t ( x )) In particular, for a 1-form α we have ( L X α ) i = X j j α i + α j i X j . More generally, since the flow ϕ t : M M is a di ff eomorphism it naturally acts on general tensor bundles of type ( p , q ), that is, ϕ * t : ( T p q ) ϕ t ( x ) M ( T p q ) x M For a general tensor field T of type ( p , q ) on M , ϕ * t T is a tensor field of type ( p , q ) defined by ( ϕ * t T ) k 1 ... k p i 1 ... i q ( x ) = ∂ϕ j 1 t ( x ) x i 1 · · · ∂ϕ j q t ( x ) x i q x k 1 ∂ϕ m 1 t ( x ) · · · x k p ∂ϕ m p t ( x ) T m 1 ... m p j 1 ... j q ( ϕ t ( x )) Definition 5.2.4 Let T be a tensor field of type ( p , q ) on M. The Lie derivative of T with respect to X is a tensor field L X T of type ( p , q ) defined by

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differential geometry w notes from teacher_Part_57 - 113...

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