So, trilinear forms are a thing: for example, if you have 3 vectors, and you want to know the volume of the parallelepiped they form, that’s a trilinear form. And that clearly has a “cubicness” to it, and you can do this for arbitrary numbers of vectors and covectors. The Riemann curvature tensor is perhaps the most significant one that has more than 2 (co)vectors involved. FWIW the dual space thing also seems likely to be important for my confusion about why phase space “volume” is 2-dimensional (even in super huge phase spaces)!
I would say that distance is bilinear in arbitrary dimension because it’s also inherently a comparison of two vectors (a vector to measure, and a “unit” vector to measure it by). Not sure if that reduces things any for you.
For me, it doesn’t feel like there’s going to be anything beyond “because comparison is important, and inherently 2-ish” for this. I do think part of why a metric is so significant is related to the dual space, but my guess is that even this will ultimately boil down to “comparison” (maybe as the concept of equality) being important.
Awesome!
So, trilinear forms are a thing: for example, if you have 3 vectors, and you want to know the volume of the parallelepiped they form, that’s a trilinear form. And that clearly has a “cubicness” to it, and you can do this for arbitrary numbers of vectors and covectors. The Riemann curvature tensor is perhaps the most significant one that has more than 2 (co)vectors involved. FWIW the dual space thing also seems likely to be important for my confusion about why phase space “volume” is 2-dimensional (even in super huge phase spaces)!
I would say that distance is bilinear in arbitrary dimension because it’s also inherently a comparison of two vectors (a vector to measure, and a “unit” vector to measure it by). Not sure if that reduces things any for you.
For me, it doesn’t feel like there’s going to be anything beyond “because comparison is important, and inherently 2-ish” for this. I do think part of why a metric is so significant is related to the dual space, but my guess is that even this will ultimately boil down to “comparison” (maybe as the concept of equality) being important.