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An analytic manifold is a manifold with analytic transition functions over some field. The most widely studied are real-analytic and complex analytic manifolds.
Analytic manifolds are studied in analytic geometry.
L. Schwarz, Lectures on complex analytic manifolds (1955) (pdf)
Jean-Pierre Serre: Analytic Manifolds, Part II Chapter III of: Lie Algebras and Lie Groups – 1964 Lectures given at Harvard University, Lecture Notes in Mathematics 1500, Springer (1992) [[doi:10.1007/978-3-540-70634-2](https://doi.org/10.1007/978-3-540-70634-2)]
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