1 Answers
In mathematics, an analytic manifold, also known as a C ω {\displaystyle C^{\omega }} manifold, is a differentiable manifold with analytic transition maps. The term usually refers to real analytic manifolds, although complex manifolds are also analytic. In algebraic geometry, analytic spaces are a generalization of analytic manifolds such that singularities are permitted.
For U ⊆ R n {\displaystyle U\subseteq \mathbb {R} ^{n}} , the space of analytic functions, C ω {\displaystyle C^{\omega }} , consists of infinitely differentiable functions f : U → R {\displaystyle f:U\to \mathbb {R} } , such that the Taylor series
T f = ∑ | α | ≥ 0 D α f α ! α {\displaystyle T_{f}=\sum _{|\alpha |\geq 0}{\frac {D^{\alpha }f}{\alpha !}}^{\alpha }}
converges to f {\displaystyle f} in a neighborhood of x 0 {\displaystyle \mathbf {x_{0}} } , for all x 0 ∈ U {\displaystyle \mathbf {x_{0}} \in U}. 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. C ∞ {\displaystyle C^{\infty }} , manifolds. 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. 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.