$M$ is an $n$-dimensional $C^\infty$ Hausdorff and path-connected manifold. Let's employ some atlas for it.
Let $p$ be some point on this manifold.
Various charts containing the point p exist. Choose some "reference chart" for the time being.
For any other chart containing $p$, one obtains a Jacobian operator of the second chart, with respect to the reference chart, at $p$.
The ability to switch between many different coordinate systems allows one to define composition between Jacobians - from where it's realized that they should form a group (not hard to prove).
You'd get groups for each reference coordinate system at the point $p$. These groups should be isomorphic (not hard to prove), so this group is definable on the tangent space itself.
For instance, this group should be $\mathrm{GL}(n, \mathbb{R})$ for the maximal atlas.
For various choices of atlases,you'd get various groups (because these groups are always definable). Call this the Jacobian group at the poin $p$, $\mathcal{J}_p$.
For each point, take the homomorphism $\mathrm{det}$from this "Jacobian group" to its determinant in $\mathbb{R}$. This will be a subgroup of $\mathbb{R}-\{0\}$. I'll call this the group $\mathrm{Det}_{p}$, for the point $p$. The determinant groups for different points may well be different (examples galore of this).
Here's a question: is there some "nice criterion" that I can employ to shrink my atlas, suitably enough, so that the resulting "determinant groups for each point" reveal properties of the manifold?
I'm expecting that I could get some information on the orientability of the manifold, were I to do this.
Disclaimer: I come from a physics background, so I'm not sure if everything's well-defined. Feel free to point out, and I'll suitably modify the question to make it properly defined.
Let's say that if $G \leq {\rm GL}_n(\Bbb R)$ is a subgroup, then an atlas $\mathfrak{A} =\{(U_\alpha, \varphi_\alpha)\}_{\alpha \in A}$ is a $G$-atlas if for all $\alpha,\beta \in A$ with $U_\alpha \cap U_\beta \neq \varnothing$, the Jacobian ${\rm d}(\varphi_\alpha \circ \varphi_\beta^{-1})_x$ is in $G$, for all $x \in \varphi_\beta[U_\alpha \cap U_\beta]$.
Since transition maps for any atlas are diffeomorphisms, every atlas is a ${\rm GL}_n(\Bbb R)$-atlas. So the interesting question is: when can we find an atlas with a smaller $G$?
the manifold is orientable if and only if it admits a ${\rm GL}_n^+(\Bbb R)$-atlas. This is also equivalent to finding an ${\rm SL}_n(\Bbb R)$-atlas.
the manifold admits a flat Riemannian metric if and only if it admits an ${\rm O}(n)$-atlas.
if $n = 2k$, the manifold has a complex structure if and only if it admits a ${\rm GL}_k(\Bbb C)$-atlas (there's a canonical embedding ${\rm GL}_k(\Bbb C)\hookrightarrow {\rm GL}_{2k}(\Bbb R)$).
if $n=2k$, the manifold admits a symplectic structure if and only if it admits a ${\rm Sp}_{2k}(\Bbb R)$-atlas.
Can you see where I'm getting at? The keyword here is "$G$-structure". Here, we're talking about integrable $G$-structures, which correspond to $G$-atlases. More generally, a $G$-structure on $M$ is a principal $G$-subbundle of the frame bundle of $M$, whose fibers are $G$-orbits of the right action of $G$ on the frame bundle (namely, the restriction of the ${\rm GL}_n(\Bbb R)$-action given by change of bases). The core idea here is that, for example, defining a Riemannian metric on the manifold is the same thing as (consistently) prescribing which bases of all tangent spaces will be orthonormal. Similar reasoning for other geometric structures.
See Marius Crainic's notes for more on the subject.