Riemannian metrics on positive definite matrices related to means

The Riemannian metric on the manifold of positive definite matrices is defined by a kernel function $\varphi$ in the form $K_D^{\varphi }(H,K)={\sum }_{i,j}\varphi ({\lambda }_i,{\lambda }_j{)}^{-1}Tr{P}_{i}H{P}_{j}K$ when ${\sum }_i{\lambda }_i{P}_i$ 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 $D\mapsto G(D)$ 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 $\varphi (x,y)=M(x,y{)}^{\theta }$ 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.