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differential geometry w notes from teacher_Part_20

# differential geometry w notes from teacher_Part_20 - 39 2.3...

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2.3. THE COTANGENT BUNDLE 39 A covector field is a section of the cotangent bundle. 2.3.1 Pull-Back of a Covector Let M and N be two smooth manifolds and n = dim M and m = dim N . Let ϕ : M N be a smooth map. The di ff erential ϕ * : T p M T ϕ ( p ) N is the linear transformation of the tangent spaces. Let x i be a local coordinate system in a local chart about p M and y α be a local coordinate system in a local chart about ϕ ( p ) N and i and α be the coordinate bases for T p M and T ϕ ( p ) N . Then the action of the di ff erential ϕ * is defined by ϕ * x j = m α = 1 y α x j y α . Let v = n i = 1 v i i . Then [ ϕ * ( v )] α = n j = 1 y α x j v j . Definition 2.3.2 The pullback ϕ * is the linear transformation of the cotangent spaces ϕ * : T * ϕ ( p ) N T * p M taking covectors at ϕ ( p ) N to covectors at p M, defined as follows. If α T * ϕ ( p ) N, then ϕ * ( α ) T * p M so that ϕ * ( α ) = α ϕ * : T p M R where α : T ϕ ( p ) N R . That is, for any vector

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differential geometry w notes from teacher_Part_20 - 39 2.3...

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