  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
$\sum_i\lambda_iP_i$ 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 $D\mapsto G(D)$ 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 $\phi(x,y)=M(x,y)^\theta$ 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.
