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 well-dened 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 p-form F * ( d ) = d ( F * ) . Proof : Direct calculation. di geom.tex; April 12, 2006; 17:59; p. 86...
View Full Document
- Spring '10