标准范数广义陀螺向量空间中的局部到整体等距：JB-锥的Thompson与Hilbert几何
Local-to-Global Isometries in Standard-Norm Generalized Gyrovector Spaces: Thompson and Hilbert Geometries of JB-Cones
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中文总结 AI 辅助
本文引入标准范数广义陀螺向量空间，证明其连通开子集间的双射陀螺度量等距可唯一延拓为全局等距，并应用于JB-锥的Thompson与Hilbert几何，得到局部到整体的延拓与分类结果。
中文摘要 AI 辅助
我们引入了标准范数广义陀螺向量空间：这类广义陀螺向量空间的范数值线具有通常的实数运算，且其固定标量映射是连续的。我们证明了在两个这样的空间的非空连通开子集之间的每个双射陀螺度量等距都唯一地延拓为一个全局双射陀螺度量等距。证明结合了陀螺中点、点反射、局部等距芽的延拓以及单值性；标量乘法的联合连续性是被推导出来的，而非假设的。该类别包含所有非零实赋范空间以及标准化的Möbius、Einstein和固有速度模型，并且在有限和可数$\ell^p$-积下是稳定的。我们还获得了有界映射空间例子，并且在自然连续性假设下，对于紧空间$K$，获得了标准范数函数空间GGV $C(K,G)$。这些产生了非Hilbert和混合例子。对于每个含单位元的JB-代数，正可逆锥承载一个单参数族的标准范数GGV结构，其陀螺度量通过幂映射与Thompson度量共轭。射影正锥有一个平行的族，基于模单位的变化范数，与Hilbert射影度量共轭。射影结构是可缩的。结合全局JB-锥等距分类，抽象定理为Thompson和Hilbert几何提供了局部到整体的延拓和分类结果。有限维欧几里得Jordan代数（包括Albert代数）的满秩密度矩阵和归一化态空间给出了具体的射影例子。
英文摘要
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.
