Riemannian geometry on positive definite matrices

The Riemannian metric on the manifold of positive definite matrices is defined by a kernel function φ in the form K D(H,K) = ∑ i,j φ(λi, λj) −1TrPiHPjK when ∑ i λiPi is the spectral decomposition of the foot point D and the Hermitian matrices H,K are tangent vectors. For such kernel metrics the tangent space has an orthogonal decomposition. The pull-back of a kernel metric under a mapping D 7→ G(D) is a kernel metric as well. Several Riemannian geometries of the literature are particular cases, for example, the Fisher-Rao metric for multivariate Gaussian distributions and the quantum Fisher information. In the paper the case φ(x, y) =M(x, y) is mostly studied when M(x, y) is a mean of the positive numbers x and y. There are results about the geodesic curves and geodesic distances. The geometric mean, the logarithmic mean and the root mean are important cases. AMS classification: 15A45; 15A48; 53B21; 53C22