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? :tongue:

 

[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!

 

[blue_raver22]

I would like to provide a response to this content by clarifying the concepts of interior product, inner product, and isomorphism.

First, let's define the interior product. The interior product, also known as the interior multiplication, is a mathematical operation that takes a vector and a differential form as inputs and produces a new differential form as output. This operation is denoted by the symbol "\i" and can be thought of as a contraction of the vector onto the differential form.

Next, let's define the inner product. The inner product is a mathematical operation that takes two vectors as inputs and produces a scalar as output. This operation is denoted by the symbol "\cdot" and is commonly used to measure the angle between two vectors or to calculate the length of a vector.

Now, let's discuss their relationship. While the inner product is a scalar quantity, the interior product is a differential form. They are related in that the interior product can be thought of as a generalization of the inner product, where the vector is "contracted" onto a differential form instead of another vector.

Additionally, the two concepts are related through the isomorphism mentioned in the Mathworld page. An isomorphism is a mathematical mapping between two structures that preserves certain properties. In this case, the isomorphism between the vector space V and its dual space V^* allows us to view the inner product as a special case of the interior product. This is because the isomorphism maps a vector v in V to a linear functional e(v) in V^*, which can then be used to contract with a differential form.

To summarize, the interior product and inner product are related but distinct concepts. The interior product is a generalization of the inner product and can be thought of as a contraction operation, while the inner product is a scalar quantity. The isomorphism between the vector space and its dual space allows us to view the inner product as a special case of the interior product.