Analytic manifold

In mathematics, an analytic manifold, also known as a 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 , the space of analytic functions, , consists of infinitely differentiable functions , such that the Taylor series

converges to in a neighborhood of , for all . 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. , 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.

ReferencesEdit

  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, Springer, 102, 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.