4
$\begingroup$

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

$\endgroup$
3
$\begingroup$

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 $1$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.