When does the tangent bundle of a manifold admit a flat connection? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T12:02:54Z http://mathoverflow.net/feeds/question/91852 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/91852/when-does-the-tangent-bundle-of-a-manifold-admit-a-flat-connection When does the tangent bundle of a manifold admit a flat connection? Tom LaGatta 2012-03-21T19:39:53Z 2012-03-22T03:36:20Z <p>Let $M$ be a smooth manifold, and let $TM$ denote its tangent bundle. Under what conditions does $TM$ admit a flat connection $\omega$?</p> <p><b>Edit:</b> 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. </p> http://mathoverflow.net/questions/91852/when-does-the-tangent-bundle-of-a-manifold-admit-a-flat-connection/91868#91868 Answer by Dmitri for When does the tangent bundle of a manifold admit a flat connection? Dmitri 2012-03-21T22:42:56Z 2012-03-21T23:28:09Z <p>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: <a href="http://www.springerlink.com/content/g6804q4u77327887/" rel="nofollow">http://www.springerlink.com/content/g6804q4u77327887/</a></p> <p>The following recent article of Goldman will be relevant <em>Milnor's seminal work on flat manifolds and bundles</em> <a href="http://arxiv.org/abs/1108.0216" rel="nofollow">http://arxiv.org/abs/1108.0216</a></p> http://mathoverflow.net/questions/91852/when-does-the-tangent-bundle-of-a-manifold-admit-a-flat-connection/91880#91880 Answer by Igor Belegradek for When does the tangent bundle of a manifold admit a flat connection? Igor Belegradek 2012-03-22T03:36:20Z 2012-03-22T03:36:20Z <p>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. </p>