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?

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?