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So I know that there are several bilinear forms on a finite-dimensional vector space that are studied a lot. For example, there are the inner products that are taught to every students, there are the symmetric nondegenerate bilinear forms that are used in pseudo-Riemannian geometry and relativity, and there are the symplectic forms that are said to be useful in classical mechanics. Their associated Lie groups are also studied a lot.

But I have not heard of some specific trilinear forms or higher order forms being studied. Why is that? Are bilinear forms special? Or are there actually some studies on some specific higher order forms that are studied but never mentioned to undergraduate students?

edm
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    $\det$ ${}{}{}{}{}$ – copper.hat Aug 26 '17 at 17:47
  • Another possible example - aren't higher derivatives of functions $\mathbb{R}^n \rightarrow \mathbb{R}^m$ sort of multilinear forms? Here's a nice explanation. – John P Aug 26 '17 at 17:57
  • Note that a bilinear form can be written as a linear form over the tensor product of the vector space. This allows to give the arguments as a non separable entity. – QuantumPotatoïd Nov 27 '21 at 19:01

2 Answers2

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Surely you have heard about the determinant on a vector space of dimension $n$ which is an alternating multilinear form of degree $n$. In fact, more generally geometers are interested in volume forms which are non-vanishing smooth differential forms of maximal degree on a manifold.

  • If $\omega$ is a symplectic form on a manifold of dimension $2n$, then $\omega^n$ is a volume form.

  • If $\alpha$ is a contact form on a manifold of dimension $2n+1$, then $\alpha\wedge(\mathrm{d}\alpha)^n$ is a volume form.

C. Falcon
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The cross-product is a bilinear map from $\mathbb{R}^3\times\mathbb{R}^3$ into $\mathbb{R}^3$. Its generalization to $\mathbb{R}^n$ is a $(n-1)$-linear map from$$\overbrace{\mathbb{R}^n\times\mathbb{R}^n\times\cdots\times\mathbb{R}^n}^{n-1\text{ times}}$$to $\mathbb{R}^n$. See this post.

José Carlos Santos
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