Geometric Mean for T-Positive Definite Tensors and Associated Riemannian Geometry

The geometric mean of two T-positive definite tensors is defined and shown to have properties that a "mean" should satisfy, including idempotence, inversion, commutativity, and transformation. The geometric mean is also characterized as the unique T-positive definite solution of an algebraic Riccati tensor equation, and can be expressed as solutions of algebraic Riccati matrix equations. Additionally, the Riemannian manifold associated with the geometric mean for T-positive definite tensors is investigated, and it is shown to be a Cartan-Hadamard-Riemannian manifold.