Geometries on the cone of positive-definite matrices derived from the power potential and their relation to the power means

We study a Riemannian metric on the cone of symmetric positive-definite matrices obtained from Hessian power potential function (1?det?(X)?)/?. give explicit expressions for geodesics and distance function, under suitable conditions. In scalar case, geodesic between two positive numbers coincides with weighted mean, while size at least it yields notion mean different ones given in literature. As ? tends to zero, converges logarithmic potential, that well-known associated matrix geometric mean; we show converge respectively.