Riemannian metrics on positive de nite matrices related to means

The Riemannian metric on the manifold of positive de nite matrices is de ned by a kernel function in the form K D(H;K) = P i;j ( i; j) 1TrPiHPjK when P i iPi 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 (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 classi cation: 15A45; 15A48; 53B21; 53C22