Riemannian geometry and matrix geometric means

The geometric mean of two positive definite matrices has been defined in several ways and studied by several authors, including Pusz and Woronowicz, and Ando. The characterizations by these authors do not readily extend to three matrices and it has been a long-standing problem to define a natural geometric mean of three positive definite matrices. In some recent papers new understanding of the geometric mean of two positive definite matrices has been achieved by identifying the geometric mean of A and B as the midpoint of the geodesic (with respect to a natural Riemannian metric) joining A and B. This suggests some natural definitions for a geometric mean of three positive definite matrices. We explain the necessary geometric background and explore the properties of some of these candidates.