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

$$T_f(𝐱)=\sum _{|\alpha |\ge 0}\frac{D^{\alpha }f({𝐱}_{\mathrm{𝟎}})}{\alpha !}(𝐱-{𝐱}_{\mathrm{𝟎}}{)}^{\alpha }$$

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

See also

• Complex manifold
• Analytic variety
• Algebraic geometry § Analytic geometry