# Do smooth manifolds admit linear atlases? [duplicate]

There is a theorem of Whitney showing that a smooth manifold can be endowed with a compatible real-analytic atlas (later, it was proven that this analytic structure is essentially unique).

I am curious, how much stronger structures can be put on a smooth manifold in a compatible way? Most importantly, is it possible to find an atlas such that its transition maps be elements of $GL(n)$ ($n$ being the dimension of the manifold)?

The only related thing that I have found is the concept of piecewise-linear manifold, which seems not to be what I am looking for.

## marked as duplicate by YCor, Alex Degtyarev, András Bátkai, Community♦Mar 16 '16 at 9:23

• If that were true, then the tangent bundle would be the vector bundle associated to a group homomorphism $\pi_1(M)\to \text{GL}(n)$. In particular, if $M$ is simply connected, then the tangent bundle is a product. Now consider the $2$-sphere, where every section of the tangent bundle intersects the zero section ("hairy ball theorem"). – Jason Starr Mar 15 '16 at 17:20
• Such a manifold would be affine, although your condition is a bit stronger. en.wikipedia.org/wiki/Affine_manifold – Ian Agol Mar 15 '16 at 17:31
• This question (by me) is similar. ​ ​ – user5810 Mar 15 '16 at 18:09
• Just to make clear, this question is more special than the one asked before. Here it is about transition maps in GL(n,R), the other question allowed tranistions in Aff(R^n). – ThiKu Mar 16 '16 at 9:28
• @ThiKu: you're right, but anyway the question should be rephrased to take into account the previous question, and emphasize if he really wishes the transition maps to be restriction of linear (and not only affine) transformations to open subsets. (The terminology "piecewise linear", which motivates the question, is a historical mistake in English, as it should be "piecewise affine".) – YCor Mar 16 '16 at 14:10

An $n$-dimensional manifold endowed with an atlas whose transition function are in $Gl(n,R)$ is called a radiant affine manifold.

An affine manifold $M$ is defined by an atlas whose coordinate changes are in $Aff(R^n)$, the universal cover $\hat M$ inherits an affine structure defined by a local diffeomorphism

$D_M:\hat M\rightarrow R^n$ which induces a representation

$h_M:\pi_1(M)\rightarrow Aff(R^n)$ defined by $h_M(g)=(L(h_M)(g),a_g)$.

Here $L(h_M)$ is the linear part of $h_M$. The representation $L(h_M)$ is the holonomy of a connection defined on $M$ whose curvature and torsion vanish.

The correspondence $g\rightarrow a_g$ is a $1$-cocycle for the linear holonomy $L(h_M)$, if its class is zero then the structure is radiant and the structure is defined by coordinates change which take their values in $Gl(n,R)$.

In dimension 3, radiant affine manifolds have been classified by Choi. To quote from the AMS Math Review: