real-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, $C^(U)$, consists of infinitely differentiable functions $f:U \to \R$, such that the Taylor series

$T_f(\mathbf) = \sum_\frac (\mathbf-\mathbf)^\alpha$

converges to $f(\mathbf)$ in a neighborhood of $\mathbf$, for all $\mathbf \in U$. 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.

See also

* Complex manifold
* Analytic variety
*

References

Structures on manifolds
Manifolds

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