Homogeneous Non Symmetric Special Kähler Geometries as Broken Isometry Metrics on Symmetric CV {K}ähler Manifolds:a new tool for r=2 CaNNs
Abstract
In this paper we prove in full detail the isomorphism between the solvable group 𝒮2,2+p, metric equivalent to the Calabi Vesentini symmetric space SO(2,2+p)/SO(2)× SO(2+p), and the solvable group 𝒮L(−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 IsoL(−1,p) of L(−1,p) we show that IsoL(−1,p)⊂ SO(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 IsoL(−1,p)/H where H=U(1)L× SO(p), the generator of U(1)L being in so(2,2+p), yet not in the canonical so(2)⊕so(2+p) subalgebra. The action of IsoL(−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.
Related
- Geometries with parallel, skew-symmetric and closed torsion
- Closed formula for Kronecker structure constants of a molecular basis of homogeneous symmetric functions
- Linking numbers and special differential characters in arithmetic hyperbolic manifolds
- Stable constant mean curvature disks with circular boundary in homogeneous three-manifolds
- Compact homogeneous complex manifolds
BibTeX
@misc{fre2026homogeneous,
title = {{Homogeneous Non Symmetric Special Kähler Geometries as Broken Isometry Metrics on Symmetric CV Kähler Manifolds:a new tool for $r=2$ CaNNs}},
author = {Pietro Fré and Mario Trigiante and Antoine Van Proeyen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23395},
eprint = {2609.23395},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23395},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Homogeneous Non Symmetric Special Kähler Geometries as Broken Isometry Metrics on Symmetric CV {K}ähler Manifolds:a new tool for r=2 CaNNs
AU - Pietro Fré
AU - Mario Trigiante
AU - Antoine Van Proeyen
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23395
UR - https://arxiv.org/abs/2609.23395
AB - In this paper we prove in full detail the isomorphism between the solvable group 𝒮_2,2+p, metric equivalent to the Calabi Vesentini symmetric space SO(2,2+p)/SO(2)× SO(2+p), and the solvable group 𝒮_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 Iso_L(−1,p) of L(−1,p) we show that Iso_L(−1,p)⊂ SO(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 Iso_L(−1,p)/H where H=U(1)_L× SO(p), the generator of U(1)_L being in so(2,2+p), yet not in the canonical so(2)⊕so(2+p) subalgebra. The action of Iso_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.
ER -