The Riemannian metric on the manifold of positive definite matrices is defined by a kernel function $phi$ in the form $K_D^phi(H,K)=sum_{i,j}phi(lambda_i,lambda_j)^{-1} Tr P_iHP_jK$ when $sum_ilambda_iP_i$ 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 $Dmapsto G(D)$ 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 $phi(x,y)=M(x,y)^theta$ 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.
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