When does the tangent bundle of a manifold admit a flat connection? - MathOverflow

When does the tangent bundle of a manifold admit a flat connection?

2012-03-21T19:39:53Z

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.

Answer by Dmitri for When does the tangent bundle of a manifold admit a flat connection?

2012-03-21T22:42:56Z

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

Answer by Igor Belegradek for When does the tangent bundle of a manifold admit a flat connection?

2012-03-22T03:36:20Z

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.