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Unformatted text preview: 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 : M N be a smooth map. The di ff erential * : T p M T ( 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 be a local coordinate system in a local chart about ( p ) N and i and be the coordinate bases for T p M and T ( p ) N . Then the action of the di ff erential * is defined by * x j ! = m X = 1 y x j y . Let v = n i = 1 v i i . Then [ * ( v )] = n X j = 1 y x j v j . Definition 2.3.2 The pullback * is the linear transformation of the cotangent spaces * : T * ( p ) N T * p M taking covectors at ( p ) N to covectors at p M, defined as follows....
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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.
- Spring '10