---
tag: D46QECMC10
kind: concept
name: Frölicher–Nijenhuis bracket
msc: [58A10, 53C15, 17B70]
sources:
  - url: https://en.wikipedia.org/wiki/Fr%C3%B6licher%E2%80%93Nijenhuis_bracket
same_as:
  wikidata: Q5506750
  wikipedia: Frölicher–Nijenhuis bracket
---

## Summary

The Frölicher–Nijenhuis bracket is a graded Lie bracket on differential forms with values in the tangent bundle of a [[smooth manifold|F45VQ47GPV]].
On vector fields, which are the forms of degree $0$, it is the usual [[Lie bracket of vector fields|68FGQEBD6J]].
It is defined through the derivations of the algebra of [[differential forms|0E3G0GRPZ6]] that commute with the [[exterior derivative|0ZHSTAG8ZY]].
It expresses integrability conditions, such as those of almost complex structures and of connections, as the vanishing of a bracket.

## Explanations

### L1 human

The Lie bracket of two vector fields measures how far their flows fail to commute.
Vector-valued forms are more general objects, such as a field of linear maps on tangent spaces.
The Frölicher–Nijenhuis bracket extends the Lie bracket to all of them in a consistent way.
For example, for an almost complex structure $J$, the bracket of $J$ with itself measures whether $J$ comes from complex coordinates.