Riemannian metrics on positive definite matrices related to means. II

On the manifold of positive definite matrices, a Riemannian metric Kϕ is associated with a positive kernel function ϕ on (0,∞)×(0,∞) by defining , where D is a foot point with the spectral decomposition D=∑iλiPi and H,K are Hermitian matrices (tangent vectors). We are concerned with the case ϕ(x,y)=M(x,y)θ where M(x,y) is a mean of scalars x,y>0. We clarify the isometric structure among such kernel metrics and discuss the convergence properties of geodesic distances and geodesic shortest curves along each isometric line of metrics. The metric corresponding to the square of the logarithmic mean shows up as the attractor of the whole metrics concerned.