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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.

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

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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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 at 17:20
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Such a manifold would be affine, although your condition is a bit stronger. en.wikipedia.org/wiki/Affine_manifold – Ian Agol Mar 15 at 17:31
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This question (by me) is similar. ​ ​ – Ricky Demer Mar 15 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 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 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 $Gl(n,R)$.

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

The main result is a decomposition theorem, which says that such a manifold admits a decomposition along finitely many disjoint totally geodesic tori or Klein bottles into two kinds of pieces. The first kind are convex radiant affine 3-manifolds, i.e. those for which the universal cover is real projectively homeomorphic to a convex domain in affine space. The second kind of piece is the suspension of a real projective surface of a special kind by a real projective automorphism. If Σ is a real projective (n−1)-manifold and ϕ is a real projective automorphism of Σ, there is a natural radial affine structure on the mapping torus of the automorphism ϕ, which is a quotient by a cyclic group of a component of the complement of the zero section of the tautological line bundle over Σ coming from its projective structure. If M is a closed manifold with a radiant affine structure, there is a natural developing map from the universal cover M˜ toRn−{0}, unique up to composition with an element of GL(n,R). The radial vector field X=∑ixi∂/∂xi on Rn−{0} pulls back by the developing map to a vector field on M˜ which by naturality covers a vector field on M. This vector field is known as the radial vector field on M. Carrière asked whether every compact radiant affine 3-manifold admits a total cross section to the radial flow. Together with some results of Barbot and the author, contained in an appendix, the decomposition theorem answers the Carrière conjecture in the positive. Together with some basic results in 3-manifold topology, this implies that every compact radiant affine 3-manifold with empty or totally geodesic boundary is homeomorphic to a Seifert space with Euler number zero, or is finitely covered by a surface bundle over a circle with fiber homeomorphic to a compact surface of Euler characteristic zero.

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