# Projectively equivalent connections

We can define projective structure on a manifold in two ways. First we can define it as a maximal atlas of charts from the open subsets of manifold to the projective space , such that transitions maps are locally the elements of the projective general linear group. Second , we can define it as a torsion-free projectively flat connection. Projectively flat connection is a connection which is projectively equivalent with a flat connection around each point of the manifold. Also , two connections are projectively equivalent when there is a closed one-form such that we can write : D'(X,Y) = D(X,Y) + F(X)Y+F(Y)X , D and D' are two connections , X and Y are two vector fields and F is our closed one-form. Why we need to define projectively equivalent connections ? What is it's interpretation and the relation to the projective space and connections?

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The key point is that a (torsion-free) connection on an $n$-manifold is projectively flat if and only if each point has a neighborhood that possesses a coordinate chart that takes the geodesics of the connection to straight lines in $\mathbb{R}^n$. Two such coordinate charts will have a projective overlap function. The other point is that a connection is projectively flat if and only if it is projectively equivalent to a flat connection. More generally, two connections are projectively equivalent if and only if they have the same (unparametrized) geodesics. These facts explain the relation. – Robert Bryant Feb 19 '13 at 18:09

Now, if there exists such an atlas, then there exists a projectively flat connection. One of the way to see it is as follows. Take a volume form $\Omega$ on the manifold (assumed oriented but the proof can be generalised for nonorineted manifolds). Now, it is an easy exercise to see that for any connection $D$ there exists the unique projectively equivalent connection $D'(X,Y) = D(X,Y) + F(X)Y+F(Y)X$ such that the volume form is parallel w.r.t. to this connection. In the charts from your atlas consider the projectively flat connection such that $\Omega$ is parallel. Because of uniqueness, it does not depend on the coordinate chart and is therefore a globally defined projectively flat connection.
P.S. You definition of projective equivalence of two connections is nonstandard: one does not require usually that the 1-form $F$ is closed. It does not affect the proof above though.