In differential geometry, a projective connection is a type of Cartan connection on a differentiable manifold.
The structure of a projective connection is modeled on the geometry of projective space, rather than the affine space corresponding to an affine connection.
Much like affine connections, projective connections also define geodesics.
However, these geodesics are not affinely parametrized.
Rather they are projectively parametrized, meaning that their preferred class of parameterizations is acted upon by the group of fractional linear transformations.
The first step in defining any Cartan connection is to consider the flat case: in which the connection corresponds to the Maurer-Cartan form on a homogeneous space.
In the projective setting, the underlying manifold
of the homogeneous space is the projective space RPn which we shall represent by homogeneous coordinates
[1] Let H be the isotropy group of the point
As matrices relative to the homogeneous basis,
Relative to the matrix representation above, the Maurer-Cartan form of G is a system of 1-forms
satisfying the structural equations (written using the Einstein summation convention):[2] A projective structure is a linear geometry on a manifold in which two nearby points are connected by a line (i.e., an unparametrized geodesic) in a unique manner.
Furthermore, an infinitesimal neighborhood of each point is equipped with a class of projective frames.
According to Cartan (1924), This is analogous to Cartan's notion of an affine connection, in which nearby points are thus connected and have an affine frame of reference which is transported from one to the other (Cartan, 1923): In modern language, a projective structure on an n-manifold M is a Cartan geometry modelled on projective space, where the latter is viewed as a homogeneous space for PSL(n+1,R).
In other words it is a PSL(n+1,R)-bundle equipped with such that the solder form induced by these data is an isomorphism.