  Several Riemannian metrics and families of Riemannian metrics were defined on
the manifold of Symmetric Positive Definite (SPD) matrices. Firstly, we
formalize a common general process to define families of metrics: the principle
of deformed metrics. We relate the recently introduced family of
alpha-Procrustes metrics to the general class of mean kernel metrics by
providing a sufficient condition under which elements of the former belongs to
the latter. Secondly, we focus on the principle of balanced bilinear forms that
we recently introduced. We give a new sufficient condition under which the
balanced bilinear form is a metric. It allows us to introduce the
Mixed-Euclidean (ME) metrics which generalize the Mixed-Power-Euclidean (MPE)
metrics. We unveal their link with the (u, v)-divergences and the ($\alpha$,
$\beta$)-divergences of information geometry and we provide an explicit formula
of the Riemann curvature tensor. We show that the sectional curvature of all ME
metrics can take negative values and we show experimentally that the sectional
curvature of all MPE metrics but the log-Euclidean, power-Euclidean and
power-affine metrics can take positive values.
