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Suppose $(M,g)$ is a real analytic Riemannian manifold (that is with coordinate charts that are real analytic functions with analytic inverses). Can we conclude that the manifold is always flat? The intuition is that analyticity forces the manifold to be rigid, but it seems strange to me that every real analytic manifold is flat. Are there simple examples of real analytic manifolds which are not flat?

Wreck it Ralph
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    Every smooth manifold admits an analytic atlas, and not every smooth manifold is flat (else curvature would be a very boring invariant) – Brevan Ellefsen Jun 26 '23 at 16:43
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    As a specific example (though basically every manifold you choose will be a counterexample), the sphere $S^1$ is simply the complex manifold $\mathbb{CP}^1$ (so is complex analytic, and thus in particular real analytic) and is not flat. – Brevan Ellefsen Jun 26 '23 at 16:53
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    Presumably you're requiring that the metric $g$ is also analytic w.r.t. the $C^\omega$ structure on $M$? Even with that, not every real analytic metric is flat. Real analyticity is, generally speaking, less rigid than complex analyticity. – Kajelad Jun 26 '23 at 19:28
  • Thanks to both. I think Brevan wanted to write $S^2$, but his example is ok. Kajelad, now you make me wonder, what about with a complex analytic structure? Do we always get Riemann tensor = 0 (or at least for some traces versions?) – Wreck it Ralph Jun 27 '23 at 12:33
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    @WreckitRalph Holomorphic Hermitian forms on holomorphic vector bundles are always flat, and as a result often fail to exist (see here). – Kajelad Jun 27 '23 at 18:34
  • Thank you very much! – Wreck it Ralph Jun 28 '23 at 12:07

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