Let M to be a compact orientable $m$-dimensional manifold. I am interested in the following comment given by a user of forum: suppose that "you have an atlas $\{(U_{\alpha},\mathbf{x}_{\alpha})\}_{\alpha \in I}$ such that for all $\alpha , \beta \in I$ with $U_{\alpha} \cap U_{\beta} \neq \varnothing$ the jacobian matrix of $\mathbf{x}_{\beta} \circ \mathbf{x}_{\alpha}^{-1}$ is constant (i.e., the determinant of change of coordinates matrix is constant)." (See 1).
We know that the affin manifolds satisfy this property. We would like to know if given a compact orientable $m$-dimensional manifold there exists an atlas of manifold that satisfies this property. (It is look false, but the question is more general, and I do not know how to solve it).
Att
It is true - every orientable manifold admits an atlas whose transition maps all have differentials with determinant equal to $1$. Here is how I like to think of it. By assumption, your manifold, $M$, has a global non-vanishing volume form, $\omega$. It is known that every volume form is integrable, that is, for every $p\in M$ there is a coordinate neighborhood $(U,\mathbf{X})$, with respect to which we have $dx^1\wedge\ldots\wedge dx^n=\omega$. Cover $M$ with such coordinate neighborhoods, and all transition maps will have differentials with determinant $1$.