基于关联的胞腔复形组合几何
Incidence-based Combinatorial Geometry on Cell Complexes
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中文总结 AI 辅助
本研究在组合网格演算框架基础上，开发基于关联的胞腔复形组合几何扩展，生成有限组合几何类比，为显式胞腔组织的几何提供组合支架并明确其发展所需数学问题。
中文摘要 AI 辅助
组合网格演算（CMC）利用组合微分形式及其上链表示，直接在胞腔复形上构建守恒律。本笔记为该框架开发了一种基于关联的几何扩展，用于处理显式胞腔组织上的方向量和局部几何结构。核心构造是由复形的关联结构内在生成的一族局部纤维：每个顶点承载一个由其关联边方向张成的向量空间，这提供了切空间的组合类比，其维数反映局部拓扑。这些纤维为基于关联的向量值、余向量值及自同态值上链定义了局部系数空间；定向传输映射用于比较相邻纤维上的附着状态，而典范 solder 形式则将纤维方向与底层胞腔复形结构关联起来，二者共同产生了传输、挠率、曲率和度量结构的有限组合类比。配对方面，评价杯积为运动学量和力类量提供了配对；由纤维度量诱导的丛 Hodge 算子关联向量值与余向量值上链；加权协变关联算子则定义了关联上链上提升次数的、经传输校正的运算。该框架不假设光滑流形、固定秩丛、胞腔层或局部系统，而是直接从胞腔复形所代表的组织中生成几何结构。所得理论为显式胞腔组织上的几何建立了组合支架，并明确了其进一步发展所需的主要数学问题，包括可容许传输类、度量相容性、Cartan 型结构方程及依赖于局部性的代数结构。
英文摘要
Combinatorial Mesh Calculus (CMC) formulates conservation laws directly on cell complexes using combinatorial differential forms and their cochain representations. This note develops an incidence-based geometric extension of that framework for directional quantities and local geometric structure on explicit cellular organisation. The central construction is a family of local fibres generated intrinsically by the incidence structure of the complex. Each vertex carries a vector space spanned by its incident edge directions, providing a combinatorial analogue of a tangent space whose dimension reflects local topology. These fibres define local coefficient spaces for incidence-based vector-, covector- and endomorphism-valued cochains. Directed transport maps compare states attached to neighbouring fibres, while a canonical solder form relates fibre directions to the underlying cell-complex structure. Together they give rise to finite combinatorial analogues of transport, torsion, curvature and metric structure. Evaluation cup products provide pairings between kinematic and force-like quantities, while bundle Hodge operators induced by the fibre metric relate vector- and covector-valued cochains, and weighted covariant incidence operators define degree-raising transport-corrected operations on incidence cochains. The framework does not assume a smooth manifold, fixed-rank bundle, cellular sheaf or local system. Instead, geometric structure is generated directly from the organisation represented by the cell complex. The resulting theory establishes a combinatorial scaffold for geometry on explicit cellular organisation and identifies the principal mathematical questions required for its further development, including admissible transport classes, metric compatibility, Cartan-type structure equations and locality-dependent algebraic structures.
