Math 2.0

Several complex variables and analytic spaces Theorem

Newlander–Nirenberg theorem

Builds on Almost complex manifoldLie bracket of vector fieldsSmooth manifoldFrobenius theoremRiemannian metric

Statement

Let (M,J)(M, J) be an almost complex manifold of dimension 2n2n, with JJ smooth. Let NJN_J be the Nijenhuis tensor of JJ. Then the following are equivalent.

  1. NJ=0N_J = 0 on MM.
  2. MM has an atlas of charts φ ⁣:U→ℂn\varphi \colon U \to \C^n whose transition maps are holomorphic and such that dφ∘J=i⋅dφd\varphi \circ J = i \cdot d\varphi on each UU.

In that case JJ is the almost complex structure of a unique complex manifold structure on MM.

Remarks

The implication from 2 to 1 is a direct computation: in holomorphic charts JJ has constant coefficients, so all the brackets in NJN_J cancel. The hard direction is from 1 to 2. When JJ is real analytic, consider the bundle of (0,1)(0,1)-vectors. The condition NJ=0N_J = 0 says that this bundle is closed under the bracket, and a complex form of the Frobenius theorem gives the charts. For smooth JJ one must solve a nonlinear system of partial differential equations of ∂ˉ\bar\partial type, which is the analytic heart of the proof.

In real dimension 22 the tensor NJN_J always vanishes, because NJ(X,JX)=0N_J(X, JX) = 0 and NJN_J is antisymmetric. So every almost complex structure on a surface is integrable. In particular, an oriented surface with a Riemannian metric, with JJ the rotation by a right angle, is a Riemann surface.

Going deeper

For a student

The theorem turns the question "is MM a complex manifold" into the vanishing of a tensor, a condition that can be checked locally. It is used to show that Kähler and many other geometric structures defined by tensors are really complex manifolds. On surfaces it contains the existence of isothermal coordinates. In higher dimension many almost complex manifolds have non-integrable structures; on S6S^6 the octonionic structure is not integrable, and whether S6S^6 admits any complex structure is an open problem.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Almost complex manifold, concept, no evidence levelAlmost complex manifo…conceptLie bracket of vector fields, concept, no evidence levelLie bracket of vector…conceptRiemannian metric, concept, no evidence levelRiemannian metricconceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptTangent bundle, concept, no evidence levelTangent bundleconceptFrobenius theorem, statement, no evidence levelFrobenius theoremstatementNewlander–Nirenberg theorem, statement, no evidence levelNewlander–Nirenberg t…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Almost complex manifold concept
  2. Lie bracket of vector fields concept
  3. Riemannian metric concept
  4. Smooth manifold concept
  5. Tangent bundle concept
  6. Frobenius theorem statement
  7. Newlander–Nirenberg theorem statement

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Further reading

  1. Web https://en.wikipedia.org/wiki/Almost_complex_manifold
About this page
TagVCT848WV1T
URIm2://VCT848WV1T
Kindstatement, theorem
WritingW0 stub
MSC32Q60, 53C15
StatusThis copy of the site has no build data. Run m2 status VCT848WV1T.
AboutAlmost complex manifold
UsesAlmost complex manifold, Lie bracket of vector fields, Smooth manifold, Frobenius theorem, Riemannian metric
Changed2 November 2026

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