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 $$U \subseteq \R^n$$, the space of analytic functions, $$C^{\omega}(U)$$, consists of infinitely differentiable functions $$f:U \to \R $$, 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.