Interior Product: Definition, Inner Product & Isomorphism

[Rasalhague]

I think I understand most of this Wikipedia page on the interior product ("not to be confused with inner product"):

http://en.wikipedia.org/wiki/Interior_product

I can't yet follow the drift of the Wolfram Mathworld page on the same subject:

http://mathworld.wolfram.com/InteriorProduct.html

But I was struck by their final remark: "An inner product on V gives an isomorphism $e:V \simeq V^*$ with the dual space $V^*$. The interior product is the composition of this isomorphism with tensor contraction."

This seems more like a description of the inner product or metric tensor than what Wikipedia calls the interior product. Wikipedia's interior product seems to be just a specific contraction, namely inputting a tangent vector into the first argument slot of a covariant alternating tensor. As far as I can see Wikipedia's definition doesn't make use of the isomorphism Mathworld refers to. Is Mathworld actually talking about (what the Wikipedia writer would call) the inner product in that final paragraph, or is Mathworld using a different definition of interior product from Wikipedia's (perhaps even one in which interior and inner products are in some sense the same thing)?

(Aside: Although Wikipedia refers to the C.A.T. as a "differential form", I first came across the interior product in the context of a tangent vector contracted with a volume element, which I gather is generally not the exterior derivative of anything, in spite of the conventional notation.)

 

[Ben Niehoff]

The usage on the Mathworld page is plain wrong. "Interior product" is commonly defined as on the Wikipedia page. It is also called the "insertion operator", because it sticks a vector into a differential form. And it is also sometimes referred to as "contraction" (but be careful here, because sometimes "contraction" is used to mean the thing that is on the Mathworld page!).

Properly speaking, the interior product is a metric-independent operation. You take an element of the tangent space, and stick it into an element of (some exterior power of) the cotangent space. Nowhere along this process do you need any isomorphism between these spaces. The interior and exterior products can always be defined on any manifold, even if there is no metric.

 

[Hurkyl]

The usage on the Mathworld page is plain wrong.

And mathematicians never use the same term to refer to similar or related concepts, right?

 

[Rasalhague]

Thanks, Ben and Hurkyl. In the context where I encountered the term interior product, it matched the Wikipedia description. But I guess any system of names that tries to set up a contrast between inner and interior is asking for trouble!