In mathematics, an analytic manifold, also known as a
ω
manifold, is a differentiable manifold with analytic transition maps.
[1] The term usually refers to real analytic manifolds, although complex manifolds are also analytic.
[2] In algebraic geometry, analytic spaces are a generalization of analytic manifolds such that singularities are permitted.
, the space of analytic functions,
, consists of infinitely differentiable functions
, such that the Taylor series
The requirement that the transition maps be analytic is significantly more restrictive than that they be infinitely differentiable; the analytic manifolds are a proper subset of the smooth, i.e.
[1] There are many similarities between the theory of analytic and smooth manifolds, but a critical difference is that analytic manifolds do not admit analytic partitions of unity, whereas smooth partitions of unity are an essential tool in the study of smooth manifolds.
[3] A fuller description of the definitions and general theory can be found at differentiable manifolds, for the real case, and at complex manifolds, for the complex case.
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