Request for sources concerning tensor densities and their covariant derivative

Question

I am studying an article (Geodesically equivalent metrics by V. Matveev) that uses covariant derivatives of tensor densities, but I am failing to find some literature that deals with how this is defined.

The standard mathematical literature concerning $\alpha$-densities is already quite scarce, and I have not found any good description on how to extend a given affine connection to them. I guess that if one would do so, the covariant differentiation to tensor densities would follow easily, considering them to be tensorial products of densities and tensors:

\begin{equation} \nabla(\rho\otimes T):=\nabla(\rho)\otimes T+\rho\otimes \nabla(T) \end{equation}

In general relativity some definitions of such extension can be found, but the ones I have found (mainly in d'Inverno's "Introducing Einstein's Relativity") are given only in their coordinate expression, without justification.

So, does someone know some relevant source on the matter? Thank you in advance.

Answer 1

You can indeed get a connection on density bundles induced by an affine connection. This can be either done by looking at the principal connection on the linear frame bundle of $M$ coming from a linear connection on $TM$ and at the induced linear connection on the density bundle viewed as an associated bundle as explained in the answer of @peek-a-boo in the comment above. A simpler way is to view density bundles as (non-integral) powers of a basic bundle. For this, on an $n$-manifold $M$, you consider the line bundle $(\Lambda^nT^*M)\otimes(\Lambda^nT^*M)=:L$. Of course, differemorphisms naturally act on this bundle and they act by multiplication with the square of a determinant, which shows that the bundle is trivial (although there is not canonical trivialization). This implies that for any $w\in\mathbb R$, you can form $L^w$ (either via transition functions or as an associated bundle to the linear frame bundle of $L$ which has structure group $\mathbb R_+$). This is just a simplified version of the construction via the linear frame bundle of $M$. There are different conventions for the weight of a density (in particular in conformal and projective differential geometry), so one has to be careful how the number $w$ is related to the weight.

Now any affine connection on $M$ induces a linear connection $\nabla$ on $L$ and locally any section of $L^w$ with $w\neq 0$ can be written as $s^w$ for a section $s$ of $L$. The induced covariant derivative on $L^w$ is then characterized by $\nabla_\xi s^w=ws^{w-1}\nabla_\xi s$. More explicitly, you can take $s$ to be locally non-vanshing (for example the square of the volume form of a pseudo-Riemannian metric) so it forms a local frame for $L$ and $\nabla s=\alpha s$ for some one-form $\alpha$ on $M$. For $w\neq 0$, $s^w$ is a local frame for $L^w$ and $\nabla s^w=w\alpha s^w$ and hence $\nabla fs^w=(df+w\alpha)s^w$ which describes the action of the connection on a general section of $L^w$. This connection can then be coupled to the one on tensor fields as described in your question.