Densities, pseudoforms, absolute differential forms and measures, differential forms, etc

Question

Apologies if this question is too basic, but I figured I first heard of most of these concepts on MO, so perhaps I can ask here.

Gelfand’s definition, copied from AlvarezPaiva [My edit, could be wrong: Let $M$ be a smooth $n$-manifold, and $k\leq n$.] A $k$-density [or perhaps a $(k,s)$-density for $s=1$] $\varphi$ on the manifold $M$ is a continuous real-valued function defined on the cone of simple (a.k.a. decomposable) tangent $k$-vectors on $M$ that is homogeneous of degree one. A $k$-density $\varphi$ is said to be smooth if for every $k$-tuple of smooth linearly independent vector fields $X_i$ $(1 \leq i \leq k)$ defined in some open set $U \subset M$, we have that the function $$ y \mapsto \varphi(X_1(y)\wedge \cdots \wedge X_k(y)) $$
is smooth in $U$.

A densitiy is called even if $\varphi(-v) = \varphi(v)$ for every simple tangent $k$-vector $v$. Likewise, we have odd $k$-densities that generalize differential $k$-forms

AlvarezPaiva also wrote later in a comment: 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 $v_1,\dots, v_k$ then the number depends only on the $k$-vector $v_1 \wedge \cdots \wedge v_k$ 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.

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Wikipedia has

[Let $V$ be an $n$-dimensional (real?) vector space.] If one wishes to define a function $\mu : V \times ... \times V \to \mathbb R$ that assigns a volume for any such parallelotope, it should satisfy the following properties:

• (absolute homogeneity) If any of the vectors $v_k$ is multiplied by $\lambda \in \mathbb R$, the volume should be multiplied by $|\lambda|$.
• If any linear combination of the vectors $v_1, ..., v_{j-1}, v_{j+1}, ..., v_n$ is added to the vector $v_j$, the volume should stay invariant.

These conditions are equivalent to the statement that $\mu$ is given by a translation-invariant measure on $V$, and they can be rephrased as $ {\displaystyle \mu (Av_{1},\ldots ,Av_{n})=\left|\det A\right|^s \mu (v_{1},\ldots ,v_{n}),\quad A\in \operatorname {GL} (V), s=1.}$ Any such mapping $\mu : V \times ... \times V \to \mathbb R$ is called a ($s$-)density [or perhaps $(n,s)$-density] on the vector space $V$. Note that if $(v_1, ..., v_n)$ is any basis for $V$, then fixing $\mu(v_1, ..., v_n)$ will fix $\mu$ entirely; it follows that the set $\operatorname{Vol}^s(V)$ of all densities on $V$ forms a one-dimensional vector space. Any n-form $\omega$ on $V$ defines a density $|\omega|$ on $V$ by $ {\displaystyle |\omega |(v_{1},\ldots ,v_{n}):=|\omega (v_{1},\ldots ,v_{n})|.}$

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The article on absolute differential forms defined on nlab: https://ncatlab.org/nlab/show/absolute+differential+form ends with

Apparently absolute p -forms (at least if continuous) are the same as even p -densities as defined by Gelfand

but the “Apparently” does not inspire much confidence. I learned about this nlab article from this MO answer, in which the comments seem to have some unresolved discussion on whether the definition is even “correct”.

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This MO question Pseudo-Differentialforms tries to understand pseudoforms, but AlvarezPaiva redirects to densities again, but then comments

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.

Conversely, this MO question on densities Why do I need densities in order to integrate on a non-orientable manifold? has the highest-scored answer containing a short story about/polemic against differential forms by John Baez: https://groups.google.com/g/sci.physics.research/c/aiMUJrOjE8A/m/jGy2N3IaajwJ, promoting instead pseudoforms!

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Moreover, in this MSE answer,

People tend to take the extension of Lebesgue theory from $\mathbb{R}^n$ to manifolds more or less for granted, so precise accounts of this can be oddly hard to find. However, a precise if terse account of Lebesgue theory on manifolds can be found in Dieudonné's Treatise of Analysis, Volume 3, Section 16.22 (especially Theorem 16.22.2 and the following discussion). Dieudonné doesn't require a Riemannian metric, but the point is that Riemannian metric gives a canonical choice of Lebesgue measure in the sense of Dieudonné, in exactly the same way that it gives a canonical volume form in the orientable case. In fact, Lebesgue measures in the sense of Dieudonné can be identified with nowhere vanishing $1$-densities, and the construction of the Riemannian measure $\lambda_g$ is really the construction of the canonical $1$-density $\lvert \mathrm{vol}_g \rvert$ associated to $g$.

so now we’re bringing in Riemannian/Lebesgue measures too!

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I understand my question is basic, but I hope one understands how easy it is to get confused with all this back and forth! So,

Question:

• can someone explain carefully all these definitions (densities, pseudoforms, absolute differential forms, relations with more “standard” concepts like measures and differential forms, etc.), their similarities, and differences (similarities and differences in different contexts as well, like smooth manifolds vs. Riemannian manifolds, etc.)?

• And for sake of concreteness, it would be nice to see the explicit calculation/computation for why the arclength element in 2D $\sqrt{(dx)^2+(dy)^2}$ (or 3D!) and surface area element in 3D ($\alpha=\sqrt{\left(d x_2 \wedge d x_3\right)^2+\left(d x_3 \wedge d x_1\right)^2+\left(d x_1 \wedge d x_2\right)^2}$ pg. 6 of this pdf) are $(k,1)$-densities/pseudoforms/absolute differential forms/whatnot.

  ((I must be not understanding something fundamental and basic, but I don't see how this absolute value/sign data --- "$L^1$ type things" --- is related to the arclength and surface area elements above which seem to be "$L^2$ type things".))

Answer 1

I am not the greatest expert on the details of this stuff, but since nobody else tried so far, let me have an attempt:

Prelude: Measures

Since you mention measures, I start with that, though this is more of a motivation for the objects mentioned, not one of them itself. If one wants to define a nice measure on a $k$-dimensional (sub-)manifold $M$, in order to integrate some function $f:M \to \mathbb{R}$, then one approach is to simply use a parametrization. Take a countable atlas $(\Phi_i:U_i \to M)_{i\in I}$, a matching partition of unity $(\psi_i: M \to [0,1])_{i \in I}$, maybe an additional fixed weight $g:M \to (0,\infty)$, or even $g:M \to \mathbb{R}$ and set your integration operator to be $$I_{\text{naive}}(f) := \sum_{i\in I} \int_{U_i} f(\Phi_i(x)) \psi_i(\Phi_i(x)) g(\Phi_i(x)) dx $$ where you can now use Riemann, Lebesgue or whatever type of integration on $\mathbb{R}^k$ you see fit.

This clearly works, but the result strongly depends on the atlas, which in differential geometry generally is a no go. We can fix this problem by changing the weight whenever we change the atlas. We just need a way to do so automatically, ideally using an object that lives only on $M$.

Gelfand's Densities

Since this is more or less a local thing, let me drop the atlas for now and just work with a single coordinate chart. As above, those can always be patched together using a partition of unity.

Consider now two parametrizations $\Phi_1: U_1 \to M$, $\Phi_2:U_2 \to M$ and a diffeomorphism $\Psi : U_1 \to U_2$, such that $\Phi_1 = \Phi_2 \circ \Psi$. Then by a standard change of variables $$\int_{U_2} f(\Phi_2(x)) g(\Phi_2(x)) dx = \int_{U_1} f(\Phi_1(x)) g(\Phi_1(x)) |\det D\Psi(x)| dx.$$ Of course it is no coincidence that the corresponding simple $k$-vectors $\partial_1 \Phi_1(x) \wedge \dots \wedge \partial_k \Phi_1(x)$ and $\partial_1 \Phi_2(y) \wedge \dots \wedge \partial_k \Phi_2(y)$ at $y= \Psi(x)$ similarly differ by a factor of $\det D\Psi(x)$.

So if we let $\tilde{g}$ be a 1-homogeneous, even function on the simple $k$-vector fields on $M$, we can now define $$I_{\tilde{g}}(f) := \int_U f(\Phi(x)) \tilde{g}(\Phi(x),\partial_1 \Phi(x) \wedge \dots \wedge \partial_k \Phi(x)) dx.$$ By the ideas above, this is now independent of the parametrisation. That is the Gelfand definition of an (even) density.

The Wikipedia-definition you gave considers the map $(v_1,\dots,v_k) \mapsto \tilde{g}(x,v_1\wedge \dots \wedge v_k)$ instead (also ignoring the base-point $x$). However once you take into account the properties of the exterior vector product $\wedge$, the two are equivalent.

The 1-homogenity here is crucial, to pull out the determinant as a factor. However, if $M$ is an oriented manifold, we can restrict ourselves to parametrisations of the same orientation and can drop the absolute value of the determinant. This allows us to also work with odd densities without issues. (Note that $\tilde{g}$ now needs to be allowed to take values in $\mathbb{R}$).

Differential forms

The second point is about simple $k$-vectors. These form a subset of the $k$-vectors which is closed under scalar multiplication, but not addition. The standard example is $e_1 \wedge e_2 + e_3 \wedge e_4$ which is not of the form $v_1 \wedge v_2$ for any vectors $v_1,v_2 \in \mathbb{R}^4$. Note also that being non-simple needs both dimension and co-dimension to be $\ge 2$, i.e. in $\mathbb{R}^3$ there are only simple $k$-vectors for any $k$ and for the question at hand, all this is only ever an issue on submanifolds.

Yet it is sometimes convenient to define $\tilde{g}$ on the linear space of all $k$-vectors instead (If they are not already the same). Now if we assume that $\tilde{g}$ is also odd, and the space where it is defined is linear, then once can further restrict oneself to $\tilde{g}$ being linear. But smooth enough linear maps on the space of $k$-vectors are just differential forms in the classical definition and indeed the definition of integration of a differential form is compatible with all the stuff above.

Intermediate concepts

Every other of the concepts stated is now somewhere between the two extremes, with densities being kind of the minimal objects required for integration and differential forms perhaps the nicest objects to work with.

The main issues seems to always be that, in order to be "even" and thus return the absolute value to the change of variables, one needs to break linearity. Pseudoforms do so by adding an extra factor $\pm 1$ that somehow detects a change in (local) orientation. Absolute forms more or less restrict multilinearity to adding only vectors that are already in the subspace spanned by the others (which always results in them spanning the same $k$-vector with the same sign and is thus more a question of defining an object on a tuple of $k$ vectors vs. on a single $k$-vector), and so on.

I think all the minor differences can be best understood by checking some examples mentioned, but I leave those as an exercise for the reader.

Coda: Riemannian metrics and other properties of the manifold

Now you ask, what has all this to do with the structure of the manifold? Actually not much. All the stuff above works on $C^1$-manifolds and can even be extended to Lipschitz functions and rectifiable sets with a bit of care, as the integrand needs only to be defined almost everywhere.

The only thing that is interesting is the question of when there is a canonical choice of density. The important case here is that of a Riemannian manifold, where the canonical density is that for which $\tilde{g}(x,v_1\wedge \dots \wedge v_k) = 1$ whenever $v_1,\dots,v_k$ is orthonormal. But of course there are other examples. One can also go backwards, if $M$ is also a measure space, then one can try and find the corresponding density.