The peripheral olfactory system analyses odorants through a large number of genetically defined information channels in the antennal lobe (AL), each associated to an olfactory receptor (OR) type. In Drosophila, these channels are combined in two major downstream areas - the mushroom body (MB) and lateral horn (LH). These areas are thought to support different aspects of olfactory information processing: the MB supports ongoing learning, while the LH mediates innate behavioral responses. Here, we provide a geometric analysis of representations in these different areas. We model odorant representations in each area by projecting affinity patterns for a panel of odorants onto the spaces spanned by LH and MB connectivity matrices. To compare these representations, we first perform an analysis of the intrinsic and shared dimensionalities between the two areas, using a novel method for measuring the shared dimensionality between two datasets based on a generalization of the participation ratio. We show that structure in MB is consistent with nearly random connectivity, as expected for an associative learning center, and that the LH displayed detailed structure shaping on odor representations. We also find that the structure contained in the MB overlaps with structure in the LH, suggesting some role for shared representations between MB and LH. For example, MB may also play a role in mediating innate behaviors. We then perform a clique topology analysis to understand the geometry of both representations. Our results indicate similar geometry in both areas, i.e. three-dimensional hyperbolic. Zhou et al. (2018) arrived at a similar result using perceptual data; our work suggests that this structure may also be present in the fly and may emerge as early as the LH and MB. Altogether, our results are in line with previous experimental results on the structure and function of representations in both olfactory areas.