Analytic manifold

In mathematics, an analytic manifold, also known as a ${\displaystyle C^{\omega }}$ manifold, is a differentiable manifold with analytic transition maps.^[1] The term usually refers to real analytic manifolds, although complex manifolds are also analytic.^[2] In algebraic geometry, analytic spaces are a generalization of analytic manifolds such that singularities are permitted.

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

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

converges to ${\displaystyle f(\mathbf {x} )}$ in a neighborhood of ${\displaystyle \mathbf {x_{0}} }$, for all ${\displaystyle \mathbf {x_{0}} \in U}$. 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. ${\displaystyle C^{\infty }}$, manifolds.^[1] 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.^[3] 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.

1. ^ ^{a b} Varadarajan, V. S. (1984), Varadarajan, V. S. (ed.), "Differentiable and Analytic Manifolds", Lie Groups, Lie Algebras, and Their Representations, Graduate Texts in Mathematics, vol. 102, Springer, pp. 1–40, doi:10.1007/978-1-4612-1126-6_1, ISBN 978-1-4612-1126-6
2. ^ Vaughn, Michael T. (2008), Introduction to Mathematical Physics, John Wiley & Sons, p. 98, ISBN 9783527618866.
3. ^ Tu, Loring W. (2011). An Introduction to Manifolds. Universitext. New York, NY: Springer New York. doi:10.1007/978-1-4419-7400-6. ISBN 978-1-4419-7399-3.

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