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edit approved Aug 23, 2015 at 16:05
Ali Taghavi
edit approved Aug 23, 2015 at 16:05
Ali Taghavi
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pavpanchekha
pavpanchekha
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Suppose I have a smooth manifold with a vectortangent bundle, and I have a connection. If this connection is curvature-free, is it guaranteed to be torsion-free? (I am not assuming a metric, just a finite-dimensional smooth manifold.)

I know that in general curvature-free connections do not exist, and that in general torsion-free connections do. But is the existence of a curvature-free connection sufficient to prove the existence of a torsion-free connection?

Suppose I have a smooth manifold with a vector bundle, and I have a connection. If this connection is curvature-free, is it guaranteed to be torsion-free? (I am not assuming a metric, just a finite-dimensional smooth manifold.)

I know that in general curvature-free connections do not exist, and that in general torsion-free connections do. But is the existence of a curvature-free connection sufficient to prove the existence of a torsion-free connection?

Suppose I have a smooth manifold with a tangent bundle, and I have a connection. If this connection is curvature-free, is it guaranteed to be torsion-free? (I am not assuming a metric, just a finite-dimensional smooth manifold.)

I know that in general curvature-free connections do not exist, and that in general torsion-free connections do. But is the existence of a curvature-free connection sufficient to prove the existence of a torsion-free connection?

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pavpanchekha
pavpanchekha
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Can a manifold have a curvature-free connection that is not torsion-free?

Suppose I have a smooth manifold with a vector bundle, and I have a connection. If this connection is curvature-free, is it guaranteed to be torsion-free? (I am not assuming a metric, just a finite-dimensional smooth manifold.)

I know that in general curvature-free connections do not exist, and that in general torsion-free connections do. But is the existence of a curvature-free connection sufficient to prove the existence of a torsion-free connection?