On Orbit Tangent Graphs for Lie Group Actions Through Hypergraph Incidence Structures and Separating Tangent Frameworks

This paper introduces a graphical framework for smooth Lie group actions based on tangent orbit interactions. In contrast with classical intersection graphs, where vertices usually represent algebraic subobjects and edges record set-theoretic intersections, the present construction uses non-trivial orbits as vertices and creates edges from common nonzero tangent directions inside the fixed ambient embedding. Starting from infinitesimal tangent spaces generated by the action, we construct Lie orbit tangent graphs and analyze their adjacency structure, connectedness, completeness, degrees and diameter estimates. To describe local and global interactions, tangent fibers, local tangent orbit cliques, tangent orbit hypergraphs and incidence structures are introduced. We further develop separating tangent paths and use them to construct neighborhood systems and tangent-separating topologies. The framework gives a unified way to encode orbit-level tangent interactions and may be useful in geometric analysis, symmetry-based dynamical systems, differential topology and mathematical physics, where orbits and infinitesimal directions describe invariant motions, constraints or symmetry-reduced configurations. Several examples are included to illustrate how Lie group actions, graph structures, hypergraphs and tangent geometry interact within the proposed scheme.