Analytic manifold
In mathematics, an analytic manifold, also known as a [math]\displaystyle{ C^\omega }[/math] 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 [math]\displaystyle{ U \subseteq \R^n }[/math], the space of analytic functions, [math]\displaystyle{ C^{\omega}(U) }[/math], consists of infinitely differentiable functions [math]\displaystyle{ f:U \to \R }[/math], such that the Taylor series
[math]\displaystyle{ T_f(\mathbf{x}) = \sum_{|\alpha| \geq 0}\frac{D^\alpha f(\mathbf{x_0})}{\alpha!} (\mathbf{x}-\mathbf{x_0})^\alpha }[/math]
converges to [math]\displaystyle{ f(\mathbf{x}) }[/math] in a neighborhood of [math]\displaystyle{ \mathbf{x_0} }[/math], for all [math]\displaystyle{ \mathbf{x_0} \in U }[/math]. 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. [math]\displaystyle{ C^\infty }[/math], 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.
See also
- Complex manifold
- Analytic variety
- Algebraic geometry § Analytic geometry
References
- ↑ 1.0 1.1 Varadarajan, V. S. (1984), Varadarajan, V. S., ed., "Differentiable and Analytic Manifolds" (in en), Lie Groups, Lie Algebras, and Their Representations, Graduate Texts in Mathematics (Springer) 102: pp. 1–40, doi:10.1007/978-1-4612-1126-6_1, ISBN 978-1-4612-1126-6
- ↑ Vaughn, Michael T. (2008), Introduction to Mathematical Physics, John Wiley & Sons, p. 98, ISBN 9783527618866, https://books.google.com/books?id=3Mnk63iqUc4C&pg=PA98.
- ↑ 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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