  In this paper, we generalize the geometric mean of two positive definite
matrices to that of third-order tensors using the notion of T-product.
Specifically, we define the geometric mean of two T-positive definite tensors
and verify several properties that "mean" should satisfy including the
idempotence and the commutative property, and so on. Moreover, it is shown that
the geometric mean is a unique T-positive definite solution of an algebraic
Riccati tensor equation and can be expressed as solutions of algebraic Riccati
matrix equations. In addition, we investigate the Riemannian manifold
associated with the geometric mean for T-positive definite tensors, considering
it as a totally geodesic embedded submanifold of the Riemannian manifold
associated with the case of matrices. It is particularly shown that the
geometric mean of two T-positive definite tensors is the midpoint of a unique
geodesic joining the tensors, and the manifold is a Cartan-Hadamard-Riemannian
manifold.
