  Weyl’s metric-independent construction of the affine structure led to the development of differential projective geometries or the geometries of paths. The interest in projective geometry is  in the paths, that is, in the continuous set of points of  the image set of curves, rather than in the possible  parameter descriptions of curves. A curve has one degree of freedom;  it depends on one parameter, and its image set or path is a  one-dimensional continuous set of points of the manifold. One  represents a curve on a manifold \(M\) as a smooth map  (i.e., \(C^{\infty})\) \(\gamma\) from some open interval  \(I = (-\varepsilon, \varepsilon)\) of the real line \(\mathbb{R}\) into \(M\). 