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Let $M$ be a smooth manifold, and let $TM$ denote its tangent bundle. Under what conditions does $TM$ admit a flat connection $\omega$?

Edit: Formerly, I asked about a flat connection on the frame bundle, but Deane Yang points out that a connection on the frame bundle is the same thing as one on the tangent bundle. I am imposing no other assumptions on the manifold other than smoothness, and I am seeking what assumptions may obstruct the existence of a flat connection.

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    $\begingroup$ Also, note that a connection on the frame bundle is the same thing as a connection on the tangent bundle itself. $\endgroup$ – Deane Yang Mar 21 '12 at 20:27
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    $\begingroup$ A flat and torsion-free connection on the tangent bundle is also called an affine structure. The Levi-Civita connection of a Riemannian metric is always torsion-free. $\endgroup$ – F. C. Mar 21 '12 at 20:55
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    $\begingroup$ Actually, I was asking whether you were putting any other restriction on the connection, not the manifold. You're not requiring the connection to be torsion free or the connection of a Riemannian or semi-Riemannian metric, right? $\endgroup$ – Deane Yang Mar 21 '12 at 21:29
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    $\begingroup$ I am almost certain that no one knows, even if we restrict to compact manifolds. $\endgroup$ – Ben McKay Mar 21 '12 at 22:09
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    $\begingroup$ One can turn this into an obstruction theory question by asking whether the map $M\to BGL(n,R)$ classifying the tangent bundle lifts to $BGL(n,R)^d$, where $GL(n,R)^d$ denotes $GL(n,R)$ with the discrete topology. Thus the obstructions lie in $H^k(M;\pi_{i-1}(F))$ where $F$ denotes the homotopy fiber. This is a really complicated space in general. $\endgroup$ – Paul Mar 21 '12 at 22:59
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The question of existence of flat connection on tangent bundles of manifolds was studied quite extensively. Milnor proved in one of his early papers that surfaces (compact without boundary) of non-zero Euler characteristic don't admit such a connection. A result of Smillie can be used to rule out existence of flat connection on tangent bundles of many even dimensional manifolds; a manifold $M^n$ that admit such a connection should satisfy the condition $|\chi(M^n)|\le \frac{||M^n||}{2^n}$, where $||M^n||$ denotes the simplicial norm of $M^n$. You can check http://www.ihes.fr/~gromov/PDF/4[35].pdf , page 229 for a short proof. Also, Smillie constructed examples of manifolds of non-zero Euler characteristics that admit flat connection on their tangent bundle: http://www.springerlink.com/content/g6804q4u77327887/

The following recent article of Goldman will be relevant Milnor's seminal work on flat manifolds and bundles http://arxiv.org/abs/1108.0216

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If a vector bundle admits a flat connection, then the rational Pontryagin classes of the tangent bundle vanish (as follows from Chern-Weil theory, see Milnor-Stasheff's "Characteristic classes", Appendix C, or Kobayashi-Nomidzu, volume 2). So in a sense most vector bundles do not admit flat connections.

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Is the connection supposed to be torsion free? Otherwise I think any parralelizable manifold admits a flat connection in the tangent bundle. If you require the connection to be torsion free than one obstruction is that the manifold has to have the Euler characteristic zero. This is a conjecture of Chern proven in dimension n=2.

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