This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: F * = 1 p ! 1 ... p ( y ( x )) dy 1 dy p = 1 p ! y 1 x i 1 y p x i p 1 ... p ( y ( x )) dx i 1 dx i p In components ( F * ) i 1 ... i p = y 1 x i 1 y p x i p 1 ... p ( y ( x )) Remark. The pullback is welldened only for covariant tensors and the pushforward is well dened only for contravariant tensors. Theorem 3.4.1 Let F : M W. The pullback F * : p W p M has the properties: 1. F * is linear. 2. For any two forms and F * ( ) = ( F * ) ( F * ) 3. F * commutes with exterior derivative. That is, for any pform F * ( d ) = d ( F * ) . Proof : Direct calculation. di geom.tex; April 12, 2006; 17:59; p. 86...
View
Full
Document
 Spring '10
 Wong
 Geometry

Click to edit the document details