---
tag: T46ZBWDPHQ
kind: concept
name: Nijenhuis tensor
msc: [53C15, 32Q60]
sources:
  - url: https://en.wikipedia.org/wiki/Almost_complex_manifold
---

## Summary

Let $J$ be a field of endomorphisms of the tangent bundle.
Its Nijenhuis tensor is a [[tensor field|WX302T93TW]] of type $(1,2)$ built from $J$ and the [[Lie bracket of vector fields|68FGQEBD6J]].
It is, up to a factor $2$, the [[Frölicher–Nijenhuis bracket|D46QECMC10]] of $J$ with itself.
For an [[almost complex structure|9SYWRGVTRS]], it measures the obstruction to integrability.
By the [[Newlander–Nirenberg theorem|VCT848WV1T]], an almost complex structure comes from a complex manifold structure exactly when its Nijenhuis tensor vanishes.

## Explanations

### L1 human

An almost complex structure lets one multiply tangent vectors by $i$ at each point.
One would like coordinates in which this multiplication is the usual one, as on $\C^n$.
The Nijenhuis tensor is a test, computed from derivatives of $J$, that tells whether such coordinates exist.
On $\C^n$ with its usual structure, the test gives zero; on the six-sphere with its standard almost complex structure from the octonions, it does not.