Homogeneous Non Symmetric Special Kähler Geometries as Broken Isometry Metrics on Symmetric CV {K}ähler Manifolds:a new tool for $r=2$ CaNNs

In this paper we prove in full detail the isomorphism between the solvable group ${\mathcal{S}}_{2,2+p}$, metric equivalent to the Calabi Vesentini symmetric space $\mathrm{S}\mathrm{O}(2,2+\mathrm{p})\mathrm{/}\mathrm{S}\mathrm{O}(2)\times \mathrm{S}\mathrm{O}(2+\mathrm{p})$, and the solvable group ${\mathcal{S}}_{\mathrm{L}(-1,p)}$ supporting the Kähler metric of the homogeneous non symmetric special manifold L$(-1,p)$. From an Alekseevskyan point of view the two spaces simply correspond to two different quadratic forms on the same solvable Lie algebra that we present and compare in detail. Furthermore considering the full group of isometries ${\mathrm{I}\mathrm{s}\mathrm{o}}_{\mathrm{L}(-1,p)}$ of L$(-1,p)$ we show that $\mathrm{I}\mathrm{s}{\mathrm{o}}_{\mathrm{L}(-1,\mathrm{p})}\subset \mathrm{S}\mathrm{O}(2,2+p)$ is a non-semisimple subgroup of the simple isometry group of Calabi-Vesentini manifolds. Altogether the Special Kähler manifold L$(-1,p)$ can be seen as a coset manifold $\mathrm{I}\mathrm{s}{\mathrm{o}}_{\mathrm{L}(-1,\mathrm{p})}\mathrm{/}\mathrm{H}$ where $\mathrm{H}={\mathrm{U}(1)}_L\times \mathrm{S}\mathrm{O}(\mathrm{p})$, the generator of ${\mathrm{U}(1)}_L$ being in $\text{so}(2,2+p)$, yet not in the canonical $\text{so}(2)\oplus \text{so}(2+p)$ subalgebra. The action of ${\mathrm{I}\mathrm{s}\mathrm{o}}_{\mathrm{L}(-1,p)}$ on the solvable coordinates of the CV manifold can be constructed directly. This identification provides a valuable tool for Cartan Neural Networks, introducing additional non linear transformations in every map from one layer to the next one of an $r=2$ Neural Network based on the CV Tits Satake universality class; in perspective, this new tool increases expressivity.