Smooth Structure

Differential Geometry · Axiom Academy

LESSON Smooth Structure on Manifolds Understanding how smooth atlases define differentiable structures on topological manifolds 1. Smooth (C ) Transition Functions When two charts overlap on a manifold, we need a way to translate between their coordinate systems. The transition map handles this translation. For two charts (U , ) and (U , ) with overlapping domains U U , the transition map is: The charts are smoothly compatible if this transition map -1 is infinitely differentiable (C ) wherever it's defined. 2. Smooth Atlas Defines Smooth Structure An atlas is a collection of charts that covers the entire manifold. A smooth atlas requires all transition maps between overlapping charts to be C . An atlas A = (U i , i ) on manifold M is a smooth atlas if: All transition maps are C (smoothly compatible) Different smooth atlases can define the same smooth structure if they are compatible with each other. To avoid ambiguity, we extend any smooth atlas to include all possible smoothly compatible charts. This creates a unique maximal atlas . Given a smooth atlas A , its maximal atlas A max contains every chart (U, ) that is smoothly compatible with all charts in A . 4. Different Smooth Structures on the Same Topological Manifold Remarkably, the same topological manifold can support multiple incompatible smooth structures. Two smooth atlases define different structures if they contain charts that are not smoothly compatible. When are smooth structures different?

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