## What manifolds can have a (non-piecewise) linear structure?

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By the definition I'm using, all manifolds are Hausdorff and second countable.
For all non-negative integers $n$, I define $B_n$ is to be `$\: \big\{\mathbf{v} \in \mathbf{R}^n : ||\mathbf{v}||<1\big\} \:$`.

For what manifolds does there exist an atlas of charts
$c : U\to B_n \:$ such that the transition maps are all locally affine?

(Replacing "locally affine" with "piecewise affine" would make the answer, by
definition, those manifolds for which there exists a piecewise linear structure.)

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Are you looking for sufficient conditions or a list of examples? – Misha Nov 18 at 13:43

## 1 Answer

If I am right, such manifolds are called affine manifolds. They are smooth manifolds together with a flat, torsion free connection. Maybe it is worth recalling Chern's conjecture that the Euler characteristic of an affine manifold should vanish.

Konstant B. and Sullivan D. in: "The Euler characteristic of an affine space form is zero, Bull. Amer. Math. Soc. 81 (1975)", no. 5, 937-938 proved this conjecture in the case of the quotient of the ordinary space ${\mathbb R}^n$ by a discrete group of affine transformations.

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2
David -- a small remark: the Chern conjecture is for compact affine manifolds; otherwise it is obviously false (take the Euclidean 3-space minus a point). – algori Nov 18 at 14:23
algori-- thank you for your remark. – David C Nov 18 at 14:44
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