Koordinons

Koordinons are a theoretical construct used to formalize local coordinate systems on differentiable manifolds. A koordinon consists of a local chart U ⊆ M and a coordinate map φ: U → R^n that assigns to each point x ∈ U its coordinate vector φ(x). In this framework, the data of a koordinon serves as a compact representation of a local coordinate frame, capturing both the domain of definition and the coordinate functions that identify points with n-tuples of numbers.

When two koordinons (U, φ) and (V, ψ) overlap (U ∩ V ≠ ∅), the transition map ψ ∘ φ^{-1}: φ(U ∩ V) → ψ(U ∩ V) encodes how coordinates change between the two frames. The Jacobian of this transition at a point describes the local linear change of coordinates. The collection of koordinons together must satisfy compatibility conditions (smooth transition maps) to define a differentiable structure on the manifold.

This viewpoint makes a koordinon analogous to a chart in an atlas, and the set of all koordinons provides a coordinon atlas. The usual requirement of compatibility ensures that the atlas endows M with a differentiable structure. In many treatments, the notion of an atlas is used directly, but the coordinon concept emphasizes the data carried by a local chart.

Examples include the standard Euclidean coordinates on R^n, where the canonical coordinon is (R^n, id_R^n). In R^2, polar coordinates (r, θ) on the open set with r > 0 form another coordinon. On the sphere, a pair of stereographic charts yields a small atlas of koordinons that cover the surface.

Notes: The term coordinon is not widely used in mainstream differential geometry and is primarily described in speculative or educational contexts as a way to discuss local coordinate frames and transitions. See also coordinate chart, atlas, manifold, Jacobian.