nLab Frölicher–Nijenhuis bracket



Differential geometry

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id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


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Suppose MM is a smooth manifold. Recall (e.g. from the article Nijenhuis–Richardson bracket) that any differential ( k + 1 ) (k+1) -form KΩ k+1(M,TM)K\in\Omega^{k+1}(M,T M) valued in the tangent bundle of MM gives rise to a graded derivation ι K\iota_K of degree kk on the de Rham algebra of differential forms on MM: on 1-forms we have ι Kω=ωK\iota_K \omega=\omega\circ K and on higher forms we extend using the Leibniz rule.


ι Kω(X 1,,X k+l)=1(k+1)!(l1)! σ(1) σω(K(Y σ(1),,Y σ(k+1)),Y σ(k+2),), \iota_K \omega(X_1,\ldots,X_{k+l}) \;=\; \frac {1} {(k+1)!(l-1)!} \sum_\sigma (-1)^\sigma \omega\big( K(Y_{\sigma(1)} ,\, \ldots ,\, Y_{\sigma(k+1)}) ,\, Y_{\sigma(k+2)} ,\,\ldots \big) \,,

where the sum is over all permutations σ\sigma \in Sym ( k + l ) Sym(k+l) and where (1) σ(-1)^\sigma denotes the sign of the permutation.

Cartan's magic formula

L X=[ι X,d]L_X=[\iota_X,d]

makes it natural to define the Lie derivative with respect to KΩ k(M,TM)K\in\Omega^k(M,\, T M):

L K=[ι K,d].L_K=[\iota_K,d].

The map LL defines an injective homomorphism of graded vector spaces from Ω(M,TM)\Omega(M,TM) to graded derivations of Ω(M)\Omega(M). Its image comprises precisely those derivations DD such that [D,d]=0[D,d]=0 and is closed under the commutator operation. Transferring the bracket to its source yields the Frölicher–Nijenhuis bracket:

L [K,L]=[L K,L L]L_{[K,L]} = [L_K,L_L]

for a uniquely defined [K,L]Ω k+l(M,TM)[K,L]\in\Omega^{k+l}(M,TM).

Classification of graded derivations of differential forms

Taken together, the Frölicher–Nijenhuis bracket and the Nijenhuis–Richardson bracket allows us to fully classify the graded derivations of the de Rham algebra of differential forms on MM:

A graded dervation DD of degree kk on Ω(M)\Omega(M) has a unique presentation of the form

D=L K+ι L,D=L_K + \iota_L,

where KΩ k(M,TM)K\in\Omega^k(M,TM), LΩ k+1(M,TM)L\in\Omega^{k+1}(M,TM).

We have L=0L=0 if and only if [D,d]=0[D,d]=0 and K=0K=0 if and only if DD vanishes on 0-forms.

Finally, the graded commutator of graded derivations can be expressed in terms of the Frölicher–Nijenhuis bracket (for KK) and the Nijenhuis–Richardson bracket (for LL):

[L K,L L]=L [K,L],[L_K,L_L]=L_{[K,L]},
[ι K,ι L]=ι [K,L] ,[\iota_K,\iota_L]=\iota_{[K,L]^\wedge},
[L K,ι L]=ι [K,L](1) klL ι LK,[L_K,\iota_L]=\iota_{[K,L]}-(-1)^{k l}L_{\iota_L K},
[ι K,L L]=L ι LK+(1) kι [L,K].[\iota_K,L_L]=L_{\iota_L K}+(-1)^k \iota_{[L,K]}.

Explicit formula

Given a smooth manifold MM and differential forms PΩ k(M,TM)P\in\Omega^k(M,TM), QΩ l(M,TM)Q\in\Omega^l(M,TM) valued in the tangent bundle TMTM of MM, their Frölicher–Nijenhuis bracket is a differential form

[P,Q]Ω k+l(M,TM)[P,Q]\in\Omega^{k+l}(M,TM)

defined by the formula

[P,Q](X 1,,X k+l)1k!l! σ(1) σ( [P(Y σ(1),,Y σ(k)),Q(Y σ(k+1),,Y σ(k+l))] lQ([P(Y σ(1),,Y σ(k)),Y σ(k+1)],Y σ(k+2),) +(1) klkP([Q(Y σ(1),,Y σ(k)),Y σ(k+1)],Y σ(k+2),) +12(1) k1klQ(P([Y σ(1),Y σ(2)],Y σ(3),),Y σ(k+2),) +12(1) (k1)lklP(Q([Y σ(1),Y σ(2)],Y σ(3),),Y σ(k+2),)), \begin{array}{l} [P,Q](X_1,\ldots,X_{k+l}) \;\coloneqq\; \frac{1}{k! l!} \sum_\sigma (-1)^\sigma \Big( & \big[ P(Y_{\sigma(1)},\ldots, Y_{\sigma(k)}), \, Q(Y_{\sigma(k+1)},\ldots,Y_{\sigma(k+l)}) \big] \\ & -l \, Q\big( [P(Y_{\sigma(1)},\ldots,Y_{\sigma(k)}) ,\, Y_{\sigma(k+1)}] ,\,Y_{\sigma(k+2)}, \ldots \big) \\ & +(-1)^{k l} k \, P\big( [Q(Y_{\sigma(1)},\ldots,Y_{\sigma(k)}) ,\, Y_{\sigma(k+1)}], Y_{\sigma(k+2)},\ldots\big) \\ & + \frac{1}{2} (-1)^{k-1} k l \, Q\big( P([Y_{\sigma(1)},Y_{\sigma(2)}],Y_{\sigma(3)}, \ldots),Y_{\sigma(k+2)},\ldots \big) \\ & + \frac{1}{2} (-1)^{(k-1)l} k l \, P\big( Q([Y_{\sigma(1)},Y_{\sigma(2)}],Y_{\sigma(3)},\ldots),Y_{\sigma(k+2)}, \ldots \big) \Big) \,, \end{array}

and (1) σ(-1)^\sigma is the sign of the permutation σ\sigma.


The Nijenhuis tensor of an almost complex structure JΩ 1(M,TM)J\in\Omega^1(M,TM) is [J,J][J,J]. The explicit formula yields

[J,J](X,Y)=2([JX,JY][X,Y]J[X,JY]J[JX,Y]). [J,J](X,Y) \;=\; 2\big( [J X, J Y] - [X,Y] - J[X, J Y] - J[J X,Y] \big).


The original definition, with an explicit formula is in Section 6 of

Further development:

Refinements for almost complex structures:

Discussion as a natural operation:

An article in Encyclopedia of Mathematics:

A textbook account: Chapter 16 of

  • Peter W. Michor, Topics in Differential Geometry, Graduate Studies in Mathematics 93 (2008). PDF.

Last revised on May 4, 2023 at 00:50:10. See the history of this page for a list of all contributions to it.