We study the cone $\mathscr{ P}_{ \text{ J}}$ of positive J-Hermitian matrices associated with an indefinite signature matrix J = $\text{ Id}_{ p,q}$. We show that the J-exponential map is bijective and use it to analyze the algebraic and geometric structure of $\mathscr{ P}_{ \text{ J}}$. Through a canonical identification with the cone of positive definite matrices, we endow $\mathscr{ P}_{ \text{ J}}$ with a natural Riemannian structure. In this setting, we define a J-geometric mean as the midpoint of geodesics and prove that it is uniquely characterized as the solution of a Riccati-type equation.
Categories: math.DG math.FA math.OA