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


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:


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.

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


  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

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  6. hi thank you for this post.

    when we insert X into a slot of say a, then don’t we need to move X so that it will be on the first slot of a? And this will require multiplying by (-1) times the number of swaps.

    Comment by circa1687 | November 15, 2015 | Reply

  7. That’s true, circa1687. Think of it like this: because of antisymmetry, the interior product doesn’t really care which slot you insert X into, since it just picks up a sign as you move it to the front. Inserting into the second slot of \alpha is the same as inserting it into the second slot of \alpha\wedge\beta, but inserting into the second slot of \beta is like inserting it into the p+2th slot of \alpha\wedge\beta, so we always pick up an extra p signs.

    Comment by John Armstrong | November 15, 2015 | Reply

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