In this paper, we consider data acquired by multimodal sensors capturing\ncomplementary aspects and features of a measured phenomenon. We focus on a\nscenario in which the measurements share mutual sources of variability but\nmight also be contaminated by other measurement-specific sources such as\ninterferences or noise. Our approach combines manifold learning, which is a\nclass of nonlinear data-driven dimension reduction methods, with the well-known\nRiemannian geometry of symmetric and positive-definite (SPD) matrices. Manifold\nlearning typically includes the spectral analysis of a kernel built from the\nmeasurements. Here, we take a different approach, utilizing the Riemannian\ngeometry of the kernels. In particular, we study the way the spectrum of the\nkernels changes along geodesic paths on the manifold of SPD matrices. We show\nthat this change enables us, in a purely unsupervised manner, to derive a\ncompact, yet informative, description of the relations between the\nmeasurements, in terms of their underlying components. Based on this result, we\npresent new algorithms for extracting the common latent components and for\nidentifying common and measurement-specific components.\n