On a smooth $n$ dimensional manifold $M$, it is possible to define a smooth scalar density as a pointwise-defined function $s$ which takes $n$ vector fields $X_1, \dots, X_n \in \Gamma(TM)$, and return a $C^\infty(M)$ function $s(X_1, \dots, X_n)$, such that for any section $A$ of $\text{End}(TM)$ (i.e. a map which associates with each point $p$ a endomorphism $A_p$ of $T_pM$ in a smooth manner),

$$ s(AX_1, \dots, AX_n) = |\det(A)| \cdot s(X_1, \dots, X_n). $$

Intuitively $s$ can be viewed as assigning a volume to each $n$ dimensional paralleliped in space. Naturally, they are the core objects which can be integrated on a manifold (differential forms allow integration on an oriented manifold). One can form a one dimensional vector bundle $\xi$ whose sections are precisely the scalar densities.

My question is whether there exists a similar construction which enables one to integrate on lower dimensional submanifolds. In particular, given a pointwise defined function $s$ which takes $k$ vector fields $X_1, \dots, X_k \in \Gamma(TM)$, and spits out a $C^\infty(M)$ function, what properties should $s$ have in order for us to consider it as a natural way of measuring the area of $k$ dimensional paralleliped's on the tangent space to $M$. One such property that should be true is that for any constants

$$ \{ a_{ij} : i,j \in \{ 1, \dots, k \} \} $$

and any vector fields $X_1, \dots X_k \in \Gamma(TM)$, we should have

$$ s \left( \sum_i a_{1i} X_i, \dots, \sum_i a_{ki} X_i \right) = |\det(a_{ij})| \cdot s(X_1, \dots, X_k). $$

Certainly, if $N$ is a $k$ dimensional submanifold, then a function $s$ with this property restricts to a scalar density for vector fields $\Gamma(TN)$, and can therefore be integrated on $N$. But this doesn't seem like the only property that is necessary, since the family of functions at each point with this property is infinite dimensional (consider the case $k = 1$ where $s$ is simply a seminorm chosen smoothly on each tangent space, and there are infinitely many seminorms). Thus it seems unlikely we can make a vector bundle whose sections are precisely the '$k$ dimensional' scalar densities. What further properties are needed?

  • $\begingroup$ If you want it to measure a $k$ dimensional parallelpiped, you would want there to be an $n-k$ dimensional kernel $K$ in the sense that if $Y\in K$ then $s(Y, \ldots) = 0$ (for arbitrary other input), no? I think with this you get back to something finite dimensional. $\endgroup$ Mar 17, 2020 at 1:53
  • $\begingroup$ Or maybe I misunderstand what you intend to do. Perhaps you intend to have an object $s$ such that given any immersion of a $k$ manifold $\phi: N \to M$ the pull-back $\phi^* s$ is a density on $N$? And you want to talk about the class of all such $s$? $\endgroup$ Mar 17, 2020 at 1:59
  • $\begingroup$ I mean that for any k vectors, $s(X_1, \dots, X_k)$ should measure the volume of the paralleliped generated by $X_i$. In particular, if the vectors are linearly independent, then we should have $s(X_1, \dots X_k) \neq 0$. In particular $s$ will not be linear it's entries, though it will be homogenous in the sense that $s(aX_1, \dots X_k) = |a| s(X_1, \dots, X_k)$. $\endgroup$ Mar 17, 2020 at 3:52
  • $\begingroup$ In terms of the second comment, essentially this is what I want, but hopefully this definition will imply that the set of all such densities are sections of a finite dimensional vector bundle - the functions satisfying the property I gave above do not satisfy this property since they form an infinite dimensional family, and therefore more needs to be assumed. $\endgroup$ Mar 17, 2020 at 4:00


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