Symmetric positive-definite (SPD) manifold
The set of C-by-C symmetric positive-definite matrices forms an open convex cone inside the vector space of symmetric matrices, and endowed with a suitable metric it is a smooth Riemannian manifold of dimension C(C+1)/2. It is not a vector space: a difference of two SPD matrices need not be SPD, and the straight Euclidean segment between two covariances can pass through non-positive-definite (invalid) matrices, so linear averaging and Euclidean distances are geometrically inconsistent with the space.
Treating covariance features as points on this manifold, rather than as flat vectors, lets classifiers respect its curvature: distances follow geodesics that stay inside the cone, and means are computed so they remain SPD. This single change of geometry underlies the robustness and transferability of Riemannian BCI.
Average two 2-by-2 covariances entrywise (Euclidean) versus along the geodesic: the Euclidean mean can have a larger determinant than either input (swelling), while the geometric mean does not.
Euclidean vs geometric averaging of covariances on the cone.
Near the boundary of the cone (singular, rank-deficient matrices) affine-invariant distances diverge, which is why keeping covariances well-conditioned via shrinkage matters before any manifold operation.