Dual-connection midpoint matrix means and their Gauss composition

Nakamura [J. Comput. Appl. Math. 131 (2001)] proved that the arithmetic–harmonic matrix iteration converges quadratically to the geometric matrix mean, the Riemannian midpoint of the affine-invariant metric on positive-definite matrices. We ask, more generally, when a Riemannian midpoint is a Gauss compound mean. A Riemannian metric and an affine connection induce three midpoint maps, for the connection, its metric dual, and the Levi–Civita connection. The Gauss composition of the first two equals the Levi–Civita midpoint exactly when the latter is invariant under one step of the iteration, and this invariance holds whenever an isometry acts as the point reflection about the Levi–Civita midpoint and exchanges the connection with its dual; for sufficiently close initial pairs it implies quadratic convergence of the iterations. Nakamura’s iteration is the model case, and the criterion extends it to a one-parameter family of matrix iterations whose Gauss composition is again the geometric matrix mean. A dually flat Hessian metric shows that duality alone does not suffice, while Euclidean metrics with a parallel cubic form provide non-flat examples in every dimension larger than one. In dimension one, the invariance is characterized completely through the Matkowski–Suto equation, and we conjecture that for Euclidean dual pairs it forces the cubic form to be constant, which we prove in dimension one and for scalar multiples of a constant cubic form.