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contraction (Definition)

Definition Let $ \omega$ be a smooth $ k$-form on a smooth manifold $ M$, and let $ \xi$ be a smooth vector field on $ M$. The contraction of $ \omega$ with $ \xi$ is the smooth $ (k-1)$-form that maps $ x\in M$ to $ \omega_x(\xi_x, \cdot)$. In other words, $ \omega$ is point-wise evaluated with $ \xi$ in the first slot. We shall denote this $ (k-1)$-form by $ \iota_\xi\omega$. If $ \omega$ is a 0-form, we set $ \iota_\xi \omega = 0$ for all $ \xi$.

Properties Let $ \omega$ and $ \xi$ be as above. Then the following properties hold:

  1. For any real number $ k$
    $\displaystyle \iota_{k \xi} \omega = k\iota_\xi \omega.$
  2. For vector fields $ \xi$ and $ \eta$
    $\displaystyle \iota_{\xi+\eta} \omega$ $\displaystyle =$ $\displaystyle \iota_\xi \omega + \iota_\eta \omega,$  
    $\displaystyle \iota_{\xi} \iota_\eta \omega$ $\displaystyle =$ $\displaystyle -\iota_\eta \iota_\xi \omega,$  
    $\displaystyle \iota_{\xi} \iota_\xi \omega$ $\displaystyle =$ $\displaystyle 0.$  

  3. Contraction is an anti-derivation [1]. If $ \omega^1$ is a $ p$-form, and $ \omega^2$ is a $ q$-form, then
    $\displaystyle \iota_\xi \big(\omega^1\wedge \omega^2\big) = (\iota_\xi \omega^1) \wedge \omega^2 + (-1)^p\ \omega^1\wedge (\iota_\xi \omega^2).$

Bibliography

1
T. Frankel, Geometry of physics, Cambridge University press, 1997.



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Cross-references: real number, properties, maps, vector field, smooth manifold, smooth
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This is version 1 of contraction, born on 2003-05-10.
Object id is 4261, canonical name is Contraction3.
Accessed 3484 times total.

Classification:
AMS MSC: 15A75 (Linear and multilinear algebra; matrix theory :: Exterior algebra, Grassmann algebras)
 58A10 (Global analysis, analysis on manifolds :: General theory of differentiable manifolds :: Differential forms)
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