The structure of the affine space is generalized by assuming that the change in a vector upon parallel transportation is given by terms containing not only a three-index symbol, but a one-index symbol as well. The relation between the affine connections and the metric tensor is established by the requirement that the length of a vector remains constant under parallel transportation. The one-index symbol is then taken to be proportional to the electromagnetic potential, the curvature tensor is derived, and its contractions are presented. The field equations are derived from an action principle which is formed from scalars derived from the curvature tensor. Maxwell's equations in curved space are obtained and gravitational field equations are obtained that differ from the Einstein equations by the presence of a term bilinear in the electromagnetic potential. It is shown that this term may represent the attractive force capable of balancing the repulsive stress of a sphere of charge.