nLab Frölicher–Nijenhuis bracket

Definition

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

Concretely,

$$\iota_K \omega(X_1,\ldots,X_{k+l}) \;=\; \frac {1} {(k+1)!(l-1)!} \sum_\sigma (-1)^\sigma \omega\big( K(Y_{\sigma(1)} ,\, \ldots ,\, Y_{\sigma(k+1)}) ,\, Y_{\sigma(k+2)} ,\,\ldots \big) \,,$$

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

Cartan's magic formula

$$L_X=[\iota_X,d]$$

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

$$L_K=[\iota_K,d].$$

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

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

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

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 $M$:

A graded dervation $D$ of degree $k$ on $\Omega(M)$ has a unique presentation of the form

$$D=L_K + \iota_L,$$

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

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

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

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

$$[\iota_K,\iota_L]=\iota_{[K,L]^\wedge},$$

$$[L_K,\iota_L]=\iota_{[K,L]}-(-1)^{k l}L_{\iota_L K},$$

$$[\iota_K,L_L]=L_{\iota_L K}+(-1)^k \iota_{[L,K]}.$$

Explicit formula

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

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

defined by the formula

$$\begin{array}{l} [P,Q](X_1,\ldots,X_{k+l}) \;\coloneqq\; \frac{1}{k! l!} \sum_\sigma (-1)^\sigma \Big( & \big[ P(Y_{\sigma(1)},\ldots, Y_{\sigma(k)}), \, Q(Y_{\sigma(k+1)},\ldots,Y_{\sigma(k+l)}) \big] \\ & -l \, Q\big( [P(Y_{\sigma(1)},\ldots,Y_{\sigma(k)}) ,\, Y_{\sigma(k+1)}] ,\,Y_{\sigma(k+2)}, \ldots \big) \\ & +(-1)^{k l} k \, P\big( [Q(Y_{\sigma(1)},\ldots,Y_{\sigma(k)}) ,\, Y_{\sigma(k+1)}], Y_{\sigma(k+2)},\ldots\big) \\ & + \frac{1}{2} (-1)^{k-1} k l \, Q\big( P([Y_{\sigma(1)},Y_{\sigma(2)}],Y_{\sigma(3)}, \ldots),Y_{\sigma(k+2)},\ldots \big) \\ & + \frac{1}{2} (-1)^{(k-1)l} k l \, P\big( Q([Y_{\sigma(1)},Y_{\sigma(2)}],Y_{\sigma(3)},\ldots),Y_{\sigma(k+2)}, \ldots \big) \Big) \,, \end{array}$$

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

Applications

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

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

Related concepts

• Schouten–Nijenhuis bracket

• Nijenhuis–Richardson bracket

• Nijenhuis tensor

References

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

• Albert Nijenhuis, Jacobi-type identities for bilinear differential concomitants of certain tensor fields. II, Indagationes Mathematicae (Proceedings) 58: (1955), 398-403. doi

Further development:

• Alfred Frölicher, Albert Nijenhuis, Theory of vector-valued differential forms. Part I. Derivations in the graded ring of differential forms, Indagationes Mathematicae (Proceedings) 59 (1956), 338–350 and 351–359. doi:10.1016/s1385-7258(56)50046-7 and doi:10.1016/s1385-7258(56)50047-9.

Refinements for almost complex structures:

• Alfred Frölicher, Albert Nijenhuis, Theory of vector-valued differential forms. Part II. Almost-complex structures, Indagationes Mathematicae (Proceedings) 61 (1958), 414–421 and 422-429. doi:10.1016/s1385-7258(58)50057-2 and doi:10.1016/s1385-7258(58)50058-4.

Discussion as a natural operation:

• Ivan Kolář, Peter Michor, Jan Slovák, p. 175 of: Natural operations in differential geometry, Springer (1993) [webpage, doi:10.1007/978-3-662-02950-3, pdf]

An article in Encyclopedia of Mathematics:

• Peter W. Michor, Frölicher-Nijenhuis bracket, HTML, PDF.

A textbook account: Chapter 16 of

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