This paper seeks to generalize a proposed model for grid cell learning in order to accommodate motion in curved space. The original model predicted grid cell motion on flat spaceas Lie group actions on the high dimensional grid cell vector. Expanding upon this idea, this thesis considers a general manifold where each covering chart is small enough to approximate flat space. Grid cell excitations are then sections of the vector bundle above the terrain manifold, and moving from one chart to the next promotes a global remapping of the grid cells, represented by a gauge transformation of the bundle space. Paths along this terrain may then be associated by an isotropic collection of connection one-forms, taking values in the Lie algebra of the motion group. The connections may be used to define grid field strength and curvature, which then may be used to analyze error in grid cell mapping due to permutations in the terrain. Experimental results are displayed using Neural Network learned weights to represent the updated grid cell firing structures.