analytic manifold

In mathematics, an analytic manifold, also known as a $C^\backslash omega$ manifold, is a differentiable manifold with analytic transition maps.{{Citation|last=Varadarajan|first=V. S.|title=Differentiable and Analytic Manifolds|date=1984|work=Lie Groups, Lie Algebras, and Their Representations|pages=1–40|editor-last=Varadarajan|editor-first=V. S.|series=Graduate Texts in Mathematics|volume=102|publisher=Springer|language=en|doi=10.1007/978-1-4612-1126-6_1|isbn=978-1-4612-1126-6}} The term usually refers to real analytic manifolds, although complex manifolds are also analytic.{{citation|last=Vaughn|first=Michael T.|title=Introduction to Mathematical Physics|url=https://books.google.com/books?id=3Mnk63iqUc4C&pg=PA98|page=98|year=2008|publisher=John Wiley & Sons|isbn=9783527618866}}. In algebraic geometry, analytic spaces are a generalization of analytic manifolds such that singularities are permitted.

For $U\; \backslash subseteq\; \backslash R^n$, the space of analytic functions, $C^\{\backslash omega\}(U)$, consists of infinitely differentiable functions $f:U\; \backslash to\; \backslash R$, such that the Taylor series

$$T\_f(\backslash mathbf\{x\})\; =\; \backslash sum\_\{|\backslash alpha|\; \backslash geq\; 0\}\backslash frac\{D^\backslash alpha\; f(\backslash mathbf\{x\_0\})\}\{\backslash alpha!\}\; (\backslash mathbf\{x\}-\backslash mathbf\{x\_0\})^\backslash alpha$$

converges to $f(\backslash mathbf\{x\})$ in a neighborhood of $\backslash mathbf\{x\_0\}$, for all $\backslash mathbf\{x\_0\}\; \backslash in\; U$. 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. $C^\backslash infty$, 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.{{Cite book|last=Tu|first=Loring W.|title=An Introduction to Manifolds|date=2011|publisher=Springer New York|isbn=978-1-4419-7399-3|series=Universitext|location=New York, NY|doi=10.1007/978-1-4419-7400-6}} 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.