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?