Nijenhuis tensor

The Nijenhuis tensor is a mathematical object from differential geometry . The tensor field is named after the mathematician Albert Nijenhuis . On the basis of the Newlander-Nirenberg theorem , one can use the Nijenhuis tensor to decide whether a manifold with an almost complex structure has a complex structure that induces the almost complex one.

definition

Let be a tensor field of rank (1,1) on a differentiable manifold , i.e. one has a linear map for each (smoothly depending on the base point) . The Nijenhuis tensor is then through ${\displaystyle A}$ ${\displaystyle M}$ ${\displaystyle x\in M}$ ${\displaystyle A_{x}\colon T_{x}M\rightarrow T_{x}M}$

${\displaystyle N_{A}(X,Y):=-A^{2}[X,Y]+A([AX,Y]+[X,AY])-[AX,AY]}$

(for vector fields ) defined tensor field of rank (1,2). The square brackets denote the Lie brackets of vector fields, i.e. the Lie derivation . ${\displaystyle X,Y}$

Newlander-Nirenberg theorem

An almost complex structure on a smooth manifold is a smooth map with the property that the restriction to the tangent space to each point is a bijective linear map that satisfies. ${\displaystyle M}$ ${\displaystyle J\colon TM\to TM}$ ${\displaystyle J_{p}:=J|_{T_{p}M}}$ ${\displaystyle p\in M}$ ${\displaystyle J_{p}^{2}=-\mathrm {id} }$

A complex manifold is automatically an almost complex one. Due to the complex structure, the tangential spaces become complex vector spaces and this defines an almost complex structure. Conversely, an almost complex manifold does not in general need to have a complex structure. But if there is an atlas with maps whose target area is a complex vector space and which are holomorphic in the sense of the almost complex structure, then this atlas is a complex atlas that induces the almost complex structure. Complex manifolds can therefore also be defined as almost complex manifolds that have a holomorphic atlas. In this case the almost complex structure is called integrable.

Newlander-Nirenberg's theorem : An almost complex structure can be integrated if and only if its Nijenhuis tensor vanishes. ${\displaystyle J:TM\rightarrow TM}$

${\displaystyle N_{J}(X,Y)=0\ \forall \ X,Y}$.

example

Every almost complex structure can be integrated on a 2-dimensional manifold . ${\displaystyle F}$ ${\displaystyle J}$

Proof: To check the vanishing of the Nijenhuis tensor at any point , it suffices to check the vanishing of the Nijenhuis tensor for two basis vectors of . As a basis you can choose and for one . Insertion into the Nijenhuis tensor there ${\displaystyle x\in F}$ ${\displaystyle \dim T_{x}F=2}$ ${\displaystyle T_{x}F}$ ${\displaystyle X}$ ${\displaystyle JX}$ ${\displaystyle X\in T_{x}F}$

${\displaystyle N_{J}(X,JX)=\left[X,JX\right]+J(\left[JX,JX\right]+\left[X,-X\right])-\left[JX,-X\right]=\left[X,JX\right]+\left[JX,X\right]=0.}$

Individual evidence

1. ↑ Kentaro Yano: Notes on My Mathematical Works . In: M. Obata (Ed.): Selected Papers of Kentaro Yano . Elsevier Science Ltd, 1982, ISBN 978-0-444-86495-6 , pp. XVIII .