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Hi, assume we have an torsionfree connection (not metric!) on an Riemannian manifold $(M, g)$. Then can I flow my connection such that it is metric? Maybe with the Ricci flow?

Regards Florian M.

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I think you need to try yourself to define more precisely what you mean by "flow my connection" and provide additional information on you've tried, before any of us can offer any useful advice. –  Deane Yang Nov 24 '10 at 14:48
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In particular, if you begin with a Lorentzian manifold (there are some topological restrictions) the associated torsion-free connection will presumably need to do something unpleasant to jump to a positive metric. $$ $$ en.wikipedia.org/wiki/Levi-Civita_connection $$ $$ en.wikipedia.org/wiki/Pseudo-Riemannian_manifold $$ $$ –  Will Jagy Nov 24 '10 at 18:22
    
The space of connections is an affine, so you can easily flow to a metric connection. As Deane Yang wrote, you should say what kind of flow you are intressted in. –  Sebastian Nov 25 '10 at 7:28
    
Maybe with the Ricci flow... Best regards Florian –  Florian Modler Dec 1 '10 at 18:13