We study the covariance geometry associated with probability measures in aGaussian concentration regime. Near a nondegenerate minimum of a smoothpotential, Laplace asymptotics identify a local Gaussian approximation, sothat the quadratic Wasserstein geometry of the concentrating family isasymptotically described by the geometry of Gaussian measures. On theGaussian manifold, the quadratic Wasserstein distance is exact and, in thecentred covariance sector, reduces to the Bures--Wasserstein metric on thecone of positive-definite matrices. The corresponding intrinsicgradient-flow structure on the centred Gaussian submanifold is thereforefinite-dimensional and takes place on covariance space. The main result of the paper concerns the internal geometry of thiscovariance sector in even dimension. Using Williamson's symplecticdecomposition, we consider the local orbit generated by independent one-modesymplectic transformations and compute the Bures--Wasserstein orthogonalcomplement of its tangent space at a Williamson normal form. We show thatthis orthogonal complement decomposes explicitly into radial directions,which change the symplectic eigenvalues, and off-diagonal directions, whichcreate cross-mode covariance couplings. Consequently, if a smooth functionalis critical along the local orbit, then its Bures--Wasserstein gradient liesin this transverse space; if the radial coefficients vanish, the gradient ispurely off-diagonal. We also record the behaviour of the Bures--Wasserstein metric near theboundary of the positive-semidefinite cone: in the classical concentrationregime, the covariance approaches the rank-deficient boundary, where themetric becomes singular in transverse directions. A brief final remarkexplains how the same covariance variables admit a quantum Gaussianinterpretation, but the main metric results of the paper are entirelyclassical.