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I'm looking for a definition of pseudo differential forms in ordinary differential geometry. However searching the web gave me nothing. There are definitions in supergeometry but that is not what I'm after.

Recently I read, that pseudo-differentialforms are the natural structure to integrate, since integration works on any kind of submanifold (orientation not required) for them, but those texts don't gave a 'clean' definition of these kind of forms.

What are pseudo-differentialforms?

Can pseudo differentialforms be defined as sections of some kind of fiber bundle? If yes that's a definition I would prefer.

Nevermind
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  • I suspect you are looking for a definition of 1-densities. – Eugene Lerman Feb 07 '13 at 15:13
  • They're not densities. They're forms with coefficients in a (flat) real line bundle. – Liviu Nicolaescu Feb 07 '13 at 15:15
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    I'm guessing you're looking for the notion of k-densities as explanined in my answer to this MO question: http://mathoverflow.net/questions/90455

    If you insist on a complicated definition they are sections of a determinant line bundle over the grassmannian bundle on manifold, but they're simple objects that we use every day like $\sqrt{dx^2 + dy^2}$

     – alvarezpaiva Feb 07 '13 at 15:33
  • By the way, here is another MO question on this topic http://mathoverflow.net/questions/99488. – alvarezpaiva Feb 07 '13 at 15:35
  • Is your definition of a density functorial? – Nevermind Feb 07 '13 at 16:01
  • What do you mean by a complicated definition? I want a definition that is natural. Those appear to me usually as the most easiest... – Nevermind Feb 07 '13 at 16:04
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    That definition is for people who do not know what they're talking about: a k density assigns a number to every k-dimensional parallelotope in the tangent space of a manifold in such a way that if the parallelotope is formed by tangent vectors $v_1$,...$v_k$, then the number depends only on the k-vector $v_1 \wedge \cdtos \wedge v_k$ and is homogeneous of degree 1 as a function of the k-vector. I don't know who came up with this definition, I learned it from Gelfand and these objects did appear in a work of his with S. Gindikin, but there are much earlier instances in the work of L.C. Young. – alvarezpaiva Feb 07 '13 at 18:02
  • A pseudoform on a manifold $M$, or a form of odd type in De Rham's terminology, is a form on with coefficients in the orientation sheaf of $M$. More precisely, if we let $\det TM$ denote the top exterior power of $TM$, then a pseudoform f degree $k$ on $M$ is a section of $(\Lambda^k T^* M)\otimes \det TM$. – Liviu Nicolaescu Feb 08 '13 at 10:17
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    @nevermind: Liviu's comments remind me that I forgot to say that densities and pseudoforms (formes impaires in de Rham) are not quite the same thing. Densities of order $k$ are basically the most general integrands that can be integrated intrinsically over any $k$-dimensional submanifold. Note that in order to integrate over a $k$-dimensiona submanifold, you do not need to know the value of the integrand on $k$-vectors that are not simple/decomposable. – alvarezpaiva Feb 08 '13 at 18:16
  • @Liviu: Why not post this as an answer? – Nevermind Feb 11 '13 at 23:42
  • @alvarezpaiva: I'll try to define 'Density' as an answer to my own question. Please comment,to see if we mean the same. – Nevermind Feb 11 '13 at 23:45

2 Answers2

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There at least two sources I am aware of.

  1. Theodore Frankel, The Geometry of Physics, Section 2.8 and 3.4.

  2. Georges De Rham, Varietes Differentiables. Formes, courants, formes harmoniques, Chap. II.

Liviu Nicolaescu
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Pseudo-Forms:

Let $M$ be a topological manifold and $PM$ the frame bundle of $M$. If $dim(M)=n$ then $PM$ is a $Gl(n)$-principal bundle.

Let $\tau: Gl(n) \to \mathbb{R} \; ; \; A \mapsto abs(det(A))$ the map, that maps any linear isomorphism $f \in Gl(n)$ to the absolute value of its determinant. This defines a left action of $Gl(n)$ on $\mathbb{R}$ by

$$\cdot: Gl(n) \times \mathbb{R} \to \mathbb{R} \; ; \; (A,x) \mapsto \tau(A)x$$

The bundle of pseudo-forms then is the associated (line) bundle

$$PM[\mathbb{R},\cdot]$$

of this action and pseudo-forms are sections of this bundle. If $M$ is smooth, this is a smooth bundle,since the action is smooth. ($\tau$ is smooth since $det(A)\neq0$ for $A\in Gl(n)$)

But this gives only pseudo-forms that behaves right in respect to integration on $dim(M)$-dimensional submanifolds. Remains the question, ow to generalize this to submanifolds of arbitrary dimension.

Nevermind
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  • Just using the absolute value here sounds a bit arbitrary to me. What's the reason for that and how to generalize? – Nevermind Feb 12 '13 at 00:02
  • Just use the bundle of $k$-frames ($k$ linearly independent tangent vectors), which is a principal $GL(k)$ bundle. The rest is the same. – Deane Yang Feb 12 '13 at 00:03
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    You want to use the absolute value, so the integral does not depend on orientation and the thing you're integrating looks more like a measure on the manifold or submanifold, rather than a differential form. in particular, it allows you to define integration on a non-orientable manifold. If you use forms without the absolute value, there are no global sections to integrate. – Deane Yang Feb 12 '13 at 00:05
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    @Nevermind. Remember the co-ordinate change formula for the integral ? What comes out is the absolute value of the determinant of the jacobian of the co-ordinate change function. If you don't have an orientation this is how things have to transform to get a sensible definition of integral. – Michael Murray Feb 12 '13 at 11:09
  • From a top-down perspective, there are only so many linear actions of [the multiplicative group of R] on a 1-dimensional vector space, so that maybe it's not surprising that this one crops up here. – Tim Campion Sep 22 '13 at 22:01