I am wondering whether there is a field of 'real analytic geometry', and if not, why not? There are branches of geometry corresponding to increasingly large sets of functions: polynomial (algebraic geometry), analytic (complex geometry), differentiable (differential geometry), continuous (topology). 'Shapes' defined by analytic functions are studied in complex geometry, but as far as I can see only complex-analytic functions: is there nothing to study about figures defined by real-analytic functions?
2026-03-25 15:22:45.1774452165
Is there a field of 'real analytic geometry'?
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Yes, there is a theory of real analytic manifolds. However, it turns out that any smooth manifold can be promoted to a real analytic manifold in a unique way. See this MO question and answers for more details.
In fact, connected closed smooth manifolds can also be realized as Nash manifolds, which are between the algebraic and analytic worlds. See wikipedia for a brief discussion, and also p.91 of this paper of Artin and Mazur. (In case the link dies at some point, this is Artin and Mazur's paper On period points, in Annals of Math., vol. 81 (1965).)