Timeline for Why do I need densities in order to integrate on a non-orientable manifold?
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19 events
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| Mar 23 at 22:09 | comment | added | Toby Bartels | @D.R. : Actually, linearity is kind of a red herring for GG densities, since $V+W$ isn't usually even decomposable just because $V$ and $W$ are; although it is when $V$ and $W$ are $n$-vectors (on a manifold of dimension $n$), since all $n$-vectors are). | |
| Mar 23 at 22:04 | comment | added | Toby Bartels | @D.R. : I'm writing an answer to your new post; but yes, Gelfand–Gindikin densities aren't required to be linear. Since they are required to be homogeneous and continuous, there is some restriction; thus odd $n$-densities end up having to be linear after all (they're the same as exterior $n$-forms). Also, even $n$-densities (equivalent to exterior pseudo-$n$-forms) have to be sort of linear ($\phi(V+W)=\phi(V)+\phi(W)$ if $\phi(V)$ and $\phi(W)$ have the same sign); so GS $s$-densities, which generalize these, have to be sort of linear too. But GG $k$-densities for $0<k<n$ have more options. | |
| Mar 23 at 21:16 | comment | added | D.R. | @TobyBartels so for the example of $\mathbb R^2$, 1-forms, at some fixed point, eating one vector, must be linear (in that its behavior eating a horizontal vector and vertical vector completely determines its behavior for all other directions of vectors), and hence can by decomposed as a linear combination of $dx,dy$; but rank 1 densities (weight 1) have no such restriction? Also it may be better to bring the conversation under my post mathoverflow.net/questions/467473/… instead of under this answer | |
| Mar 23 at 15:01 | comment | added | Toby Bartels | @D.R. : I mixed up the sense in which a density must be homogeneous of degree $1$. In general, it only needs to be positive-homogenous: $\phi(cV)=c\phi(V)$ (where $V$ is a decomposable multivector) for $c\ge0$. If it's even ($\phi(-V)=\phi(V)$), then it's absolute-homogeneous ($\phi(cV)={|c|}\phi(V)$ for any $c$); in fact, it's absolute-homogeneous iff it's both positive-homogeneous and even. But if it's odd ($\phi(-V)=-\phi(V)$), then it's just-plain homogeneous ($\phi(cV)=c\phi(V)$ for any $c$); in fact, it's homogeneous iff it's positive-homogenous and odd. Both elements above are even. | |
| Mar 23 at 14:52 | comment | added | Toby Bartels | @D.R. : The 3D surface area element takes a decomposable $2$-vector $v\wedge w$ and returns the area of the parallelogram spanned by $v$ and $w$. This is really a function of the $2$-vector (rather than only a function of the pair of vectors) because the area spanned by $cv$ and $w$ is the same as the are spanned by $v$ and $cw$. In fact, it's absolute-homogeneous of degree $1$ because this area is $|c|$ times the area of the parallelogram spanned by $v$ and $w$. It's smooth because, again, it's constant on points. It's even too. | |
| Mar 23 at 14:46 | comment | added | Toby Bartels | @D.R. : Yes, Wikipedia is only doing $n$-densities. (It does mention general $s$-densities, but this is the Guillemin–Sternberg $s$, not the Gelfand–Gindikin $n$.) To define $đs=\sqrt{\mathrm dx^2+\mathrm dy^2}$ as an operation on tangent vectors on $\mathbb R^2$, interpret $\mathrm dx$ as the $x$-component of the vector and $\mathrm dy$ as the $y$-component; in other words, the operation is the standard norm $\|{\cdot}\|$. It's smooth because, as a function of points, it's constant. It's absolute-homogeneous of degree $1$ because $\|{cv}\|=|c|^1\|v\|$. It's even because $\|{-v}\|=\|v\|$. | |
| Mar 21 at 17:40 | comment | added | D.R. | I’m sorry for the basic question, but could you do some basic concrete examples/computations of why the arclength element in 2D $\sqrt{(dx)^2+(dy)^2}$ and surface area element in 3D (pg. 6 of math.berkeley.edu/~wodzicki/H185.S11/podrecznik/2forms.pdf) are $k$-densities for $k=1,2$ (or perhaps following the advice of the comments, maybe $(k,s)$-densities for $k=1,2$ and $s=1$)? And to double check, your definition is the same as on Wikipedia en.wikipedia.org/wiki/Density_on_a_manifold? Or is it more general since you can take $k$ less than the dimension of the manifold $n$? | |
| Oct 6, 2022 at 11:05 | history | edited | Glorfindel | CC BY-SA 4.0 |
broken link fixed, cf. https://math.meta.stackexchange.com/a/34713/228959
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| Jul 18, 2016 at 19:31 | comment | added | alvarezpaiva | @AlexShpilkin, thanks! I had forgotten where it was. | |
| Jul 10, 2016 at 2:51 | comment | added | Alex Shpilkin | For the benefit of people who might want to find the Gelfand and Gindikin reference, I believe it is “Nonlocal inversion formulas in real integral geometry”, Funct. Anal. Its Appl. (1977) 11, 173–179, originally in Russian in Функц. анализ и его прил. (1977) 11(3), 12–19. | |
| Mar 5, 2013 at 10:06 | comment | added | alvarezpaiva | A curious fact is that Guillemin and Sternberg dedicate their book to Gelfand (from whose work they learned integral geometry, I assume) and then they define $\alpha$-densities without saying their notation "nicely clashes" with Gelfand's ;-) | |
| Mar 1, 2013 at 21:53 | comment | added | Toby Bartels | The point being, that it might be good to warn people about the conflict when you use the term, that's all. | |
| Mar 1, 2013 at 21:52 | comment | added | Toby Bartels | When I first read your answer at mathoverflow.net/questions/99488/… claiming that the arclength element in a Riemannian manifold is a 1-density, I thought that this was wrong, since I knew the notation mentioned by Eugene. (In that notation, a 1-density is the same thing a top-rank pseudoform, so I took your answer at first to mean that ds is a pseudoform on the curve seen as a 1-dimensional manifold in its own right, which is not exactly wrong but also not good enough.) | |
| Feb 14, 2013 at 20:57 | comment | added | alvarezpaiva | It's not my terminology. It is Gelfand and Gindikin's, I believe, and I think they introduced it before Guillemin and Sternberg used 1/2 densities. | |
| Feb 12, 2013 at 18:00 | comment | added | Eugene Lerman | Your terminology of "k-densities" nicely clashes with 1/2-densities used in geometric quantization. There are really two indices attached to densities: one is the number of vectors they eat. The other deals with how they transform, that is, a character of GL(n), which can be identified with a nonzero complex number. | |
| Mar 10, 2012 at 13:22 | history | edited | alvarezpaiva | CC BY-SA 3.0 |
added 35 characters in body
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| Mar 9, 2012 at 16:12 | comment | added | Deane Yang | I would just like to add a word of endorsement for anything written by Juan Carlo Alvarez-Paiva. | |
| Mar 9, 2012 at 15:52 | history | edited | alvarezpaiva | CC BY-SA 3.0 |
added 34 characters in body
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| Mar 9, 2012 at 15:46 | history | answered | alvarezpaiva | CC BY-SA 3.0 |