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I noticed that we had no entry density, so I very briefly created one. While cross-linking it, I noticed that at volume form there is related discussion re “pseudo-volume forms”. Maybe somebody here would enjoy to add a bit more glue? (I won’t at the moment.)
As written in volume form, the difference between a volume pseudoform (in the axiomatic sense) and a density (of weight $1$) is that the former must be positive definite (or at least positive semidefinite, but mostly positive definite).
I added something to density about the physical meaning of the term.
As written in volume form, the difference between a volume pseudoform (in the axiomatic sense) and a density (of weight 1) is that the former must be positive definite
Not sure what this is the reaction to. But if you assume a Riemannian metric in the first place, then one speaks of just the volume form anyway. One speaks of choices of volume forms rather in the absence of a Riemannian metric, in which case there is no notion of positivity.
I have edited the beginning of the Definition-section at volume form now to read as follows:
For $X$ a general smooth manifold, a volume form on $X$ is a non-vanishing density (of rank 1) on $X$. If $X$ is oriented then this is equivalently a non-vanishing section of the canonical bundle of $X$, hence an everywhere non-vanishing differential n-form on $X$, for $n$ the dimension of $X$.
More specifically still, for $(X,g)$ an oriented (pseudo)-Riemannian manifold of dimension $n$, its volume form $vol_g \in \Omega^n(X)$ is the differential form of degree $n$ which measures the volume as seen by the metric $g$ in that it is characterized by any of the following equivalent statements.
I added something to density about the physical meaning of the term.
Thanks, tha’s nice!
Oh, now I see what you mean by “positive”!
is positive (meaning that its integral on any open submanifold is nonnegative)
Okay, sure.
Not sure what this is the reaction to.
Just the relationship between the two articles.
But if you assume a Riemannian metric in the first place, then one speaks of just the volume form anyway.
Yes, that is (or was) at volume form first, and the more general axiomatic version comes under Generalisations. (Although you also need an orientation to get the volume form as an untwisted $n$-form rather than as a pseudoform.)
in the absence of a Riemannian metric, in which case there is no notion of positivity.
On the contrary, the metric is irrelevant to positivity. It's the orientation (regardless of metric) that determines positivity of $n$-forms; and positivity of pseudo-$n$-forms aka $1$-densities is (as you noticed in #5) inherent. (What the metric does is to specify two nondegenerate $n$-forms^{1}, but it doesn't tell you which is positive and which is negative.)
Actually, it specifies $2^c$ nondegenerate $n$-forms, where $c$ is the number of components of the manifold; and if you start with a degenerate metric, then you get $2^c$ continuous $n$-forms, where now $c$ is the number of components of the open submanifold where the metric is nondegenerate. ↩
There became some duplication between the section on unoriented manifolds and what came before, so I just put all of the unoriented stuff in from the beginning. Then I added a bit about degenerate volume forms.
Shouldn’t det be composed with the absolute value homomorphism to get densities as opposed to top differential forms?
Otherwise, how is det^s even defined if s is not integer and det<0?
I changed the representation so that it matched the definition of 1-densities in the provided reference (Nicole Berline, Ezra Getzler, Michele Vergne, Heat kernels and Dirac operators, Springer (2004)). Also added the prefix “1” to the word “density” in order to explicitly distinguish it from the other use of the word “density” (meaning an untwisted n-form, see differential form. This also fixes Dimitri’s concern which he expressed in his last post.
Alfredo Álvarez
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