# Bundles of relative scalars

Let $$M$$ be a smooth manifold with an atlas $$\{(U_i,\psi_i)\}$$. If $$x\in U_i\cap U_j$$, let $$a_{ji}(x)$$ denote the Jacobian matrix at $$\psi_i(x)$$ of the smooth map $$\psi_j^{-1}\circ \psi_i:\psi_i(U_i\cap U_j)\to \psi_j(U_i\cap U_j)$$. Then $$a_{ji}$$ is a smooth map $$U_i\cap U_j\to GL(n,\Bbb R)$$ and satisfies the cocycle condition $$a_{ki}=a_{kj}a_{ji}$$.

Now define $$g_{ji}=a_{ji}^m$$, where $$m$$ is a fixed positive integer, and let $$E\to M$$ be a smooth vector bundle that has the transition functions $$\{g_{ji}\}$$. Then $$E$$ is called the the bundle of relative scalars over $$M$$ of weight $$m$$, according to Steenrod's The Topology of Fibre Bundles, section 6.

I see that if $$m=1$$, then $$E\to M$$ is the tangent bundle of $$M$$. But for $$m>1$$, what is $$E$$? I mean, I haven't seen these kind of definitions in textbooks about smooth manifolds, for example, Lee's Introduction to Smooth Manifolds. But maybe these bundles have another familiar names nowadays, I hope.

I think you're misinterpreting Steenrod's definition. (Admittedly, his exposition is hard to follow on this point.) What he calls the bundle of relative scalars of weight $$\boldsymbol w$$ is the rank-$$1$$ vector bundle with transition functions $$\det(a_{ij})^w$$, where $$(a_{ij})$$ are the transition functions for the tangent bundle as you defined them above.
Modern mathematicians usually call this the bundle of densities of weight $$\boldsymbol w$$. See my answer to this question for a detailed explanation.