The issue of resolving manifold ambiguity in subspace-based direction-finding systems using planar substrates is investigated in this paper. Manifold ambiguity is often referred to a situation when subspaced-based direction-finding techniques fail to correctly identify a unique set of directions-of-arrival due to the linear dependence amongst array manifold vectors. Various attempts have been proposed to suppress or minimize the presence of this undesirable situation. The recent work proposed in [1] offers an effective method to uniquely identify a correct set of directions by using flat medium substrates to place at the frontend of antenna elements. As the plane wave propagates through the substrate, the change of direction and the corresponding propagation delay will occur. This effectively, breaks down the linear dependence amongst the manifold vectors. In this work, further investigation on the use of planar substrates to resolve ambiguity is presented from the differential geometry perspective, where the analysis will assess the array geometry to find the sensors that cause the presence of ambiguity and examine how the method can completely eliminate "all" manifoldly ambiguous sets with respect to the given AGS. A general procedure will be provided, where the description on the total number of required planar substrates together with their allocated positions shall be given. Copyright © 2013 by the Institute of Electrical and Electronic Engineers, Inc.