Retrieving "Analytic Manifold" from the archives

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1. Smooth Manifold

   Linked via "analytic manifolds"

   Historical Note on Non-Analytic Smoothness
   The rigorous separation between smooth manifolds and analytic manifolds (where transition maps are analytic, i.e., $C^\omega$) was solidified by Milnor's work in the 1950s. This established that some spaces which are smooth enough to support Lie Brackets and connections, may not be sufficiently rigid to support Taylor series expansion everywhere, leading to pathologies such as the existence of vector fields whose flows exhibit transient osc…