Analytic manifold

In mathematics, an analytic manifold, also known as a $C^{\omega }$ 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 $U\subseteq \mathbb {R} ^{n}$, the space of analytic functions, $C^{\omega }(U)$, consists of infinitely differentiable functions $f:U\to \mathbb {R}$, such that the Taylor series

$T_{f}(\mathbf {x} )=\sum _{|\alpha |\geq 0}{\frac {D^{\alpha }f(\mathbf {x_{0}} )}{\alpha !}}(\mathbf {x} -\mathbf {x_{0}} )^{\alpha }$

converges to $f(\mathbf {x} )$ in a neighborhood of $\mathbf {x_{0}}$, for all $\mathbf {x_{0}} \in U$. 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. $C^{\infty }$, 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.