Local-to-Global Isometries in Standard-Norm Generalized Gyrovector Spaces: Thompson and Hilbert Geometries of JB-Cones

Abstract

We introduce standard-norm generalized gyrovector spaces: generalized gyrovector spaces whose norm-value line has the ordinary real operations and whose fixed scalar maps are continuous. We prove that every bijective gyrometric isometry between nonempty connected open subsets of two such spaces extends uniquely to a global bijective gyrometric isometry. The proof combines gyromidpoints, point reflections, continuation of local isometry germs, and monodromy; joint continuity of scalar multiplication is derived, rather than assumed. The class includes all nonzero real normed spaces and standardized Möbius, Einstein, and proper-velocity models, and is stable under finite and countable $\ell^p$-products. We also obtain bounded mapping-space examples and, under natural continuity assumptions, standard-norm function-space GGVs $C(K,G)$ for compact $K$. These yield non-Hilbertian and hybrid examples. For every unital JB-algebra, the positive invertible cone carries a one-parameter family of standard-norm GGV structures whose gyrometrics are conjugate to Thompson's metric by power maps. The projective positive cone has a parallel family, based on the variation norm modulo the unit, conjugate to Hilbert's projective metric. The projective structures are contractible. Together with global JB-cone isometry classifications, the abstract theorem yields local-to-global extension and classification results for Thompson and Hilbert geometries. Full-rank density matrices and normalized state spaces of finite-dimensional Euclidean Jordan algebras, including the Albert algebra, give concrete projective examples.