---
tag: 9SYWRGVTRS
kind: concept
name: Almost complex manifold
msc: [53C15, 32Q60]
specializes: [F45VQ47GPV]
sources:
  - url: https://en.wikipedia.org/wiki/Almost_complex_manifold
same_as:
  wikidata: Q2995691
  wikipedia: Almost complex manifold
---

## Summary

An almost complex manifold is a smooth manifold whose tangent spaces carry a smoothly varying multiplication by $i$.
Every complex manifold is one, but the converse fails in general.
The [[Newlander–Nirenberg theorem|VCT848WV1T]] says exactly when an almost complex structure comes from complex coordinates.
Almost complex structures are basic in Hermitian, Kähler and symplectic geometry.

## Explanations

### L1 human

In the complex plane, multiplying by $i$ turns each arrow by a quarter turn.
An almost complex structure gives such a quarter turn $J$ on every tangent plane of a manifold, with $J$ applied twice giving minus the identity.
On $\R^2$ the rule $J(a, b) = (-b, a)$ is the quarter turn.
Only manifolds of even dimension can carry such a structure.