It was shown by \textit{A. G. Walker} [Convegno Internaz. Geometria Differenz., Venice/Italy 1953, 64-70 (1954; Zbl 0056.154)] that a torsion-free affine connection on a manifold canonically determines a pseudo-Riemannian metric on the cotangent bundle, called the Riemann- extension of the affine connection. In this paper the author gives an intrinsic definition of the Riemann-extension and shows that by making use of this pseudo-Riemannian metric it is possible to define an affine immersion of manifolds in affine differential geometry without making a suitable choice of normal planes.