Log-Euclidean Lie Groups
We develop a self-contained theory of log-Euclidean Lie groups: smooth manifolds diffeomorphic to finite-dimensional vector spaces, equipped with the pullback of a constant Euclidean metric. This framework encompasses symmetric positive-definite (SPD) matrices S+(n) and full-rank correlation matrices Cor+(n), and explains why many seemingly different log-Euclidean constructions yield the same Riemannian geometry. We provide explicit Riemannian isometries (and Lie group isomorphisms) linking several log-Euclidean metrics on SPD and correlation matrix manifolds, and we characterize quotient log-Euclidean metrics in a principal-bundle setting. Finally, using the diagonal correction map underlying the off-log parametrization, we construct an explicit log-Euclidean metric on S+(n) for which the standard inclusion i\,: Cor+(n) $\rightarrow$ S+(n) becomes an isometric (indeed, totally geodesic) embedding, yielding closed-form formulas for geodesics and orthogonal decompositions in adapted coordinates. The nested isometric embeddings constructed here also provide a simple solution to the comparison of matrices of different dimensions in the log-Euclidean setting: SPD or correlation matrices may be transported to a common dimension via explicit maps while preserving all intrinsic distances.