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Added some links for those unfamiliar with the terminology.
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Igor Khavkine
Igor Khavkine
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It is well known that one can specify a complex structurecomplex structure on a real $C^\infty$ manifold in two equivalent ways: an atlas with holomorphic transition functions between charts and an integrable almost complex structureintegrable almost complex structure. One of the directions is straightforward. In the other direction, the celebrated Newlander–Nirenberg theoremNewlander–Nirenberg theorem states that an integrable almost complex structure induces a holomorphic atlas.

For real analytic manifoldsreal analytic manifolds, I know only the atlas method (transition functions between charts are real analytic). My question: does there exist for real analytic manifolds an analog of an almost complex structure, an integrability condition and an analog of the Newlander–Nirenberg theorem?

At the moment, my suspicion is that the right analog should be an almost CR structureCR structure and a corresponding integrability condition (whatever those are). But unfortunately I've not really seen this written anywhere.

It is well known that one can specify a complex structure on a real $C^\infty$ manifold in two equivalent ways: an atlas with holomorphic transition functions between charts and an integrable almost complex structure. One of the directions is straightforward. In the other direction, the celebrated Newlander–Nirenberg theorem states that an integrable almost complex structure induces a holomorphic atlas.

For real analytic manifolds, I know only the atlas method (transition functions between charts are real analytic). My question: does there exist for real analytic manifolds an analog of an almost complex structure, an integrability condition and an analog of the Newlander–Nirenberg theorem?

At the moment, my suspicion is that the right analog should be an almost CR structure and a corresponding integrability condition (whatever those are). But unfortunately I've not really seen this written anywhere.

It is well known that one can specify a complex structure on a real $C^\infty$ manifold in two equivalent ways: an atlas with holomorphic transition functions between charts and an integrable almost complex structure. One of the directions is straightforward. In the other direction, the celebrated Newlander–Nirenberg theorem states that an integrable almost complex structure induces a holomorphic atlas.

For real analytic manifolds, I know only the atlas method (transition functions between charts are real analytic). My question: does there exist for real analytic manifolds an analog of an almost complex structure, an integrability condition and an analog of the Newlander–Nirenberg theorem?

At the moment, my suspicion is that the right analog should be an almost CR structure and a corresponding integrability condition (whatever those are). But unfortunately I've not really seen this written anywhere.

fixed spelling of Nirenberg; removed misspelled (overly specific) new tag (if you feel strongly about the new tag, please use 'newlander-nirenberg' or, better, 'newlander-nirenberg-theor'); added appropriate top-level tag
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Ricardo Andrade
Ricardo Andrade
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It is well known that one can specify a complex structure on a real $C^\infty$ manifold in two equivalent ways: an atlas with holomorphic transition functions between charts and an integrable almost complex structure. One of the directions is straightforward. In the other direction, the celebrated Newlander-NierenbergNewlander–Nirenberg theorem states that an integrable almost complex structure induces a holomorphic atlas.

For real analytic manifolds, I know only the atlas method (transition functions between charts are real analytic). My question: does there exist for real analytic manifolds an analog of an almost complex structure, an integrability condition and an analog of the Newlander-NierenbergNewlander–Nirenberg theorem?

At the moment, my suspicion is that the right analog should be an almost CR structure and a corresponding integrability condition (whatever those are). But unfortunately I've not really seen this written anywhere.

It is well known that one can specify a complex structure on a real $C^\infty$ manifold in two equivalent ways: an atlas with holomorphic transition functions between charts and an integrable almost complex structure. One of the directions is straightforward. In the other direction, the celebrated Newlander-Nierenberg theorem states that an integrable almost complex structure induces a holomorphic atlas.

For real analytic manifolds, I know only the atlas method (transition functions between charts are real analytic). My question: does there exist for real analytic manifolds an analog of an almost complex structure, an integrability condition and an analog of the Newlander-Nierenberg theorem?

At the moment, my suspicion is that the right analog should be an almost CR structure and a corresponding integrability condition (whatever those are). But unfortunately I've not really seen this written anywhere.

It is well known that one can specify a complex structure on a real $C^\infty$ manifold in two equivalent ways: an atlas with holomorphic transition functions between charts and an integrable almost complex structure. One of the directions is straightforward. In the other direction, the celebrated Newlander–Nirenberg theorem states that an integrable almost complex structure induces a holomorphic atlas.

For real analytic manifolds, I know only the atlas method (transition functions between charts are real analytic). My question: does there exist for real analytic manifolds an analog of an almost complex structure, an integrability condition and an analog of the Newlander–Nirenberg theorem?

At the moment, my suspicion is that the right analog should be an almost CR structure and a corresponding integrability condition (whatever those are). But unfortunately I've not really seen this written anywhere.

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Igor Khavkine
Igor Khavkine
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Analog of Newlander–Nirenberg theorem for real analytic manifolds

It is well known that one can specify a complex structure on a real $C^\infty$ manifold in two equivalent ways: an atlas with holomorphic transition functions between charts and an integrable almost complex structure. One of the directions is straightforward. In the other direction, the celebrated Newlander-Nierenberg theorem states that an integrable almost complex structure induces a holomorphic atlas.

For real analytic manifolds, I know only the atlas method (transition functions between charts are real analytic). My question: does there exist for real analytic manifolds an analog of an almost complex structure, an integrability condition and an analog of the Newlander-Nierenberg theorem?

At the moment, my suspicion is that the right analog should be an almost CR structure and a corresponding integrability condition (whatever those are). But unfortunately I've not really seen this written anywhere.