It is the homogeneous space PGL(n+1)/H where H is the isotropy group of a line.
The curved analog of projective space is a manifold in which the notion of a geodesic makes sense, but for which there are no preferred parametrizations on those geodesics.
Broadly speaking, projective geometry refers to the study of manifolds with this kind of connection.
Topologically, it is the n-sphere, but there is no notion of length defined on it, just of angle between curves.
Equivalently, this geometry is described as an equivalence class of Riemannian metrics on the sphere (called a conformal class).