The Unapologetic Mathematician

Mathematics for the interested outsider

The Interior Product

We have yet another operation on the algebra \Omega(M) of differential forms: the “interior product”. Given a vector field X\in\mathfrak{X}(M) and a k-form \omega\in\Omega^k(M), the interior product \iota_X(\omega) is the k-1-form defined by

\displaystyle\left[\iota_X\omega\right](X_1,\dots,X_{k-1})=\omega(X,X_1,\dots,X_{k-1})

That is, we just take the vector field X and stick it into the first “slot” of a k-form. We extend this to functions by just defining \iota_Xf=0.

Two interior products anticommute: \iota_X\iota_Y=-\iota_Y\iota_X, which follows easily from the antisymmetry of differential forms. Each one is also clearly linear, and in fact is also a graded derivation of \Omega(M) with degree -1:

\displaystyle\iota_X(\alpha\wedge\beta)=(\iota_X\alpha)\wedge\beta+(-1)^{-p}\alpha\wedge(\iota_X\beta)

where p is the degree of \alpha. This can be shown by reducing to the case where \alpha and \beta are wedge products of 1-forms, but rather than go through all that tedious calculation we can think about it like this: sticking X into a slot of \alpha\wedge\beta means either sticking it into a slot of \alpha or into one of \beta. In the first case we just get \iota_x\alpha, while in the second we have to “move the slot” through all of \alpha, which incurs a sign of (-1)^p=(-1)^{-p}, as asserted.

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July 26, 2011 - Posted by | Differential Topology, Topology

5 Comments »

  1. Why is this post in the “point-set topology” category?

    Comment by Andrei | July 26, 2011 | Reply

  2. An errant click on a very tightly-spaced list.

    Comment by John Armstrong | July 26, 2011 | Reply

  3. [...] turns out that there is a fantastic relationship between the interior product, the exterior derivative, and the Lie [...]

    Pingback by Cartan’s Formula « The Unapologetic Mathematician | July 26, 2011 | Reply

  4. [...] along the projection to get a “time” vector field on the cylinder. Then we use the interior product to assert that and [...]

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  5. [...] and applying it to the vector itself: . We also take the canonical volume form , and we use the interior product to define the -form [...]

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