Transitiemaps

Transitiemaps, more commonly referred to as transition maps or coordinate transition functions, are fundamental objects in differential geometry and topology. They describe how to translate coordinates between overlapping local descriptions of a space, such as a manifold.

Definition and role

Consider a manifold M equipped with two charts (U, φ) and (V, ψ) where U and V are open in M and U ∩ V ≠ ∅. The transition map is the composition φ ∘ ψ^{-1}, defined on ψ(U ∩ V) and taking values in φ(U ∩ V). It provides the rule for converting a point’s coordinates from the ψ-chart to the φ-chart on the region where the charts overlap. Transition maps encode how local data glue together to form a global geometric object.

Regularity and structure

The regularity of transition maps determines the differentiable structure of the manifold. For a topological manifold, transition maps on overlaps are required to be homeomorphisms. For a smooth (C^r) manifold, they must be C^r-diffeomorphisms. Complex manifolds require holomorphic transition maps. The collection of charts with the property that all pairwise transition maps have the specified regularity constitutes an atlas; a maximal atlas contains all charts compatible with that regularity.

Examples and significance

On the circle S^1, using two overlapping coordinate charts, the transition map on the overlap is a smooth, invertible function between open intervals. On a sphere S^2, the standard latitude-longitude charts have smooth transition maps on their overlap regions. Transition maps ensure that local coordinate descriptions yield a consistent global geometric object, enabling calculus and analysis to be performed in local coordinates while preserving global structure. See also atlas, chart, smooth manifold, and cocycle conditions.