Notes on Ehresmann’s connections -Part 2, Frölicher–Nijenhuis bracket

In this note I shall introduce the Frölicher-Nijenhuis bracket – a powerful tool, especially when dealing with connections on fibre bundles.
Let  (E,\pi,M) be a fibre bundle. In the previous note we have seen that the connection form  \omega is a one-form on   E with values in  TE (in fact, in values in  VE\subset TE.) This leads us to a more general subject of natural operations on tangent valued-forms.

Let therefore, for now,  E be any differentiable manifold, and let us consider the space of tangent valued forms on  E. Let  \Omega (E)=\oplus_{i=0}^n \Omega^{i}(E) be the graded algebra of ordinary differential forms on  E. Then the (graded) space  \Omega (E,TE) of tangent-valued forms can be written as

 \Omega(E,TE)=\Omega(E)\otimes_E T(E).

In the following we will use the capital indices  A,B,C\ldots to refer to local charts  x^A of  E.

The elements of  \Omega^{0}(E,TE) are simply vector fields  \xi^A on  E. The elements of  \Omega^{1}(E,TE) are tangent-valued one-forms. If \Phi is such a form, then \Phi assigns to each tangent vector \xi\in T_pE another tangent vector  \Phi(\xi)\in T_pE. In coordinates it is represented as  \Phi^A_B. In general, an element of  \Omega^{r}(E,TE) is represented by  \Phi^A_{B_1,\ldots B_r}, antisymmetric in indices  B_1,\ldots B_r, or

 \Phi= \frac{1}{r!}\,\Phi^A_{B_1,\ldots B_r}\,dx^{B_1}\wedge\ldots\wedge dx^{B_r}\,\partial_A.

In the following it will be convenient to introduce the following notation:


We will write

 \Phi=\sum \Phi^A_B\,d^B\otimes \partial_A,

where  B=(1\leq B_1 < B_2 < \ldots < B_r\leq\mathrm{dim }\,E),  d^B=dx^{B_1}\wedge\ldots\wedge dx^{B^r} and \partial_A=\frac{\partial}{\partial x^A}.

Operations on differential forms

Frölicher-Nijenhuis (FN) bracket will assign to any two tangent valued forms \Phi\in \Omega^{r}(E,TE),\,\Psi\in \Omega^{s}(E,TE) another form  [\Phi,\Psi]\in \Omega^{(r+s)}(E,TE).

We will see (in the following posts) that the curvature of a connection can be simply expressed in terms of the FN bracket of the connection form with itself. Also Bianchi identities for the curvature will follow immediately from this definition and from the properties of the FN bracket. But first let us recall the three fundamental operations on differential forms  \Phi\in\Omega(E). These are: exterior derivative, Lie derivative, insertion. Exterior derivative  d:\omega\mapsto d\omega maps  \Omega^{r}(E) to  \Omega^{(r+1)}(E) according to the formula

 d\Phi(X_0,\ldots ,X_r)=\sum_{i=0}^r(-1)^iX_i(\Phi(X_0,\ldots,\hat{X_i},\ldots ,X_r)).

Given a vector field  X\in\mathfrak{X} (E) the insertion operator  i_X maps  \Omega^{r}(E) to  \Omega^{(r-1)}(E) according to the formula


Lie derivative  \mathfrak{L}_X maps  \Omega^{r}(E) into itself, the definition being


These three operations on number-valued differential forms have the following important properties (note: the commutators are graded commutators):

  • d(\omega\wedge\eta)=d\omega\wedge\eta+(-1)^{|\omega|}\omega\wedge d\eta
  •  i_X(\omega\wedge\eta)=i_X\omega\wedge\eta+(-1)^{|\omega|}\omega\wedge i_X\eta
  •  \mathfrak{L}_X(\omega\wedge\eta)=\mathfrak{L}_X\omega\wedge\eta+(-1)^{|\omega|}\omega\wedge \mathfrak{L}_X \eta
  •  d^2=0
  •   [i_X,i_Y]=i_Xi_Y+i_Yi_X=0
  •  [\mathfrak{L}_X,d]=\mathfrak{L}_X\circ d+d\circ\mathfrak{L}_X=0
  •  [i_X,d]=i_X\circ d+d\circ\ i_X=\mathfrak{L}_X
  •  [\mathfrak{L}_X,\mathfrak{L}_Y]=\mathfrak{L}_X\circ\mathfrak{L}_Y-\mathfrak{L}_Y\circ\mathfrak{L}_X=\,\mathfrak{L}_{[X,Y]}
  •  [\mathfrak{L}_X,i_Y]=\mathfrak{L}_X i_Y-i_Y\mathfrak{L}_X=i_{[X,Y]}
  • If f is a function on E, then \mathfrak{L}_{fX}\omega=f\mathfrak{L}_X\omega+df\wedge i_X\omega

Frölicher–Nijenhuis bracket

Frölicher-Nijenhuis bracket associates to each pair of tangent-valued forms on  E the third tangent-valued form. Because of its bilinearity it is enough to define it on simple tensors. For \alpha\in\Omega^{r}(E),\,\beta\in\Omega^{s}(E), and for  X,Y\in \mathfrak{X}(E) it is defined by the following formula:

     \begin{eqnarray*} [\alpha\otimes X,\,\beta\otimes Y]&=&(\alpha\wedge\beta)\otimes [X,Y]\\&+&(\alpha\wedge\mathfrak{L}_X\beta)\otimes Y\\&-&(\mathfrak{L}_Y\alpha\wedge\beta)\otimes X\\&+&(-1)^r(d\alpha\wedge i_X\beta)\otimes Y\\&+&(-1)^r(i_Y\alpha\wedge d\beta)\otimes X\end{eqnarray*}

In coordinates the Frölicher-Nijenhuis bracket is given by the following explicit expression:

     \begin{eqnarray*} [\Phi,\Psi]&=&\left(\Phi^C_{B_1\ldots B_r}\partial_C\Psi^A_{B_{r+1}\ldots B_{r+s}}\right.\\ &-&(-1)^{rs}\Psi^C_{B_1\ldots B_s}\partial_C\Phi^A_{B_{s+1}\ldots B_{r+s}}\\ & -&r\Phi^A_{B_1\ldots B_{r-1} C}\partial_{B_r}\Psi^C_{B_{r+1}\ldots B_{r+s}}\\ &+&(-1)^{rs}s\Psi^A_{CB_{1}\ldots B_{s-1}}\partial_{B_{s}}\Phi^C_{B_{s+1}\ldots B_{r+s}}\left.\right)\,d^B\otimes\partial_A\end{eqnarray*}

If we want to write the sam formula in a more explicit form, then antisymmetrizations and combinatorial factors enter:

     \begin{align*} &[\Phi,\Psi](X_1,\dots X_{r+s})=\\ &=\frac{1}{r!s!}\sum_\sigma (-1)^\sigma [\Phi(X_{\sigma 1}\ldots X_{\sigma r}),\Psi(X_{\sigma(r+1)}\ldots X_{\sigma(r+s)})] \\ &+(-1)^r\left( \frac{1}{r!(s-1)!}\sum_\sigma (-1)^\sigma\Psi([X_{\sigma1},\Phi(X_{\sigma2},\ldots ,X_{\sigma (r+1)})],X_{\sigma (r+2)},\ldots)\right.\\ &\left.-\frac{1}{(r-1)!(s-1)!2!}\sum_\sigma(-1)^\sigma\Psi(\Phi([X_{\sigma 1},X_{\sigma 2}],X_{\sigma 3},\ldots),X_{\sigma (r+2)},\ldots) \right)\\ &-(-1)^{rs+s}\left(\frac{1}{(r-1)!s!}\sum_\sigma (-1)^\sigma\Phi([X_{\sigma 1},\Psi(X_{\sigma 2},\ldots ,X_{\sigma (s+1)})],X_{\sigma (s+2)},\ldots)\right .\\ &-\left.\frac{1}{(r-1)!(s-1)!2!}\sum_\sigma(-1)^\sigma\Phi(\Psi([X_{\sigma 1},X_{\sigma 2}],X_{\sigma 3},\ldots ),,X_{\sigma (s+2)},\ldots )\right) \end{align*}

As an exercise let us compute  [id_E,\beta\otimes Y], where  id_E=dx^A\otimes\partial_A is the identity tangent-valued form (the natural soldering form), with coordinates \delta^A_B. Thus we set  r=1. The first term gives

 [dx^A\otimes\partial_A,\Psi]^A_{A_1\ldots A_{r+1}}=\delta^B_{A_1}\partial_B \Psi^A_{A_2\ldots A_{s+1}}=\partial^A_{A_1}\Psi_{A_2\ldots A_{s+1}}.

The second and fourth terms vanishs because  \delta^A_B are constant.

The third is

 -\delta^A_B\partial_{A_1}\Psi^B_{A_2\ldots A_{s+1}}=-\partial_{A_1}\Psi^A_{A_2\ldots A_{s+1}}

It follows that  [dx^A\otimes\partial_A,\Psi]=0 – therefore  id_E commutes with every tangent valued form.

The most important properties of the FN bracket are the following ones

  •  [\Phi,\Psi]=-(-1)^{|\Phi|\,|\Psi|}\,[\Psi,\Phi]\quad – graded antisymmetry
  •  [\Phi_1,[\Phi_2,\Phi_3]]=[[\Phi_1,\Phi_2],\Phi_3]+(-1)^{|\Phi_1|\,\|\Phi_2|}[\Phi_2,[\Phi_1,\Phi_3]] – Jacobi identity

Suppose now  X is a vector field. It can be also considered as a tangent-valued 0-form:  X=1\otimes X. We then find that

 [X,\beta\otimes Y]=\beta\otimes [X,Y]+\mathfrak{L}_X\beta\otimes Y=\mathfrak{L}_X(\beta\otimes Y).

Therefore for vector fields the FN bracket reduces to an ordinary Lie derivative.

Another insight into the nature of the FN bracket comes from considering tangent-valued forms as operators on diferential forms. First of all, for any  \Phi \in \Omega^{r+1}(E,TE) we can define a graded derivation  i_\Phi:\,\Omega^{s}E\rightarrow \Omega^{r+s} E by the formula:

     \begin{align*} (i_\Phi\, \omega)&(X_1,\ldots,X_{r+s})=\\ &\frac{1}{(r+1)!(s-1)!}\sum_{\sigma\in S_{r+s}}(-1)^\sigma\,\omega(\Phi(X_{\sigma 1},\ldots ,X_{\sigma (r+1)}),X_{\sigma (r+2)},\ldots) \end{align*}

Now I have a problem. In references [1] and [2] we have the following local formulas:

Screenshot from: Michor, Topics in Differential Geometry

Screenshot from: Kolar, Michor, Slovak,  Natural Operations in Differential Geometry

In our notation this would be:

 i_\Phi\omega=\sum\,\Phi^A_{B_1\ldots B_{r+1}}\,\omega_{AB_{r+2}\ldots B_{r+s}}\,d^B

But I am not able to reproduce this result. It seems to me that the factor s is missing in front of the expression on the RHS. I will return to this enigma at the end of this note.

Remark. Notice that in these formulas we assume  \Phi \in \Omega^{r+1}(E,TE) and not in  \Omega^{r}(E,TE)

Then we define the Lie derivative  \mathfrak{L}_\Phi of a differential form \omega\in\Omega^s E along a tangent-valued form  \Phi\in \Omega^r(E,TE) by the formula

 \mathfrak{L}_\Phi\, \omega =i_\Phi\,d\omega-di_\Phi\,\omega


    \begin{align*} \mathfrak{L}_\Phi\,\omega=&\sum\,\left(\Phi^A_{B_1\ldots B_r}\partial_A\,\omega_{B_{r+1}\ldots B_{r+s}}\right.\\ &\left.+(-1)^r(\partial_{B_1}\Phi^A_{B_2\ldots B_{r+1}})\,\omega_{AB_{r+2}\ldots B_{r+s}}\right) d^B \end{align*}


    \begin{align*} ( &\mathfrak{L}_{\Phi}\,\omega)(X_1,\ldots ,X_{r+s})=\\ &=\frac{1}{r!s!}\,\sum_\sigma\,(-1)^\sigma \mathfrak{L}_{\Phi (X_{\sigma 1},\ldots ,X_{\sigma r})}(\omega(X_{\sigma (r+1)},\ldots,X_{\sigma (r+s)}))\\ &+(-1)^r\left(\frac{1}{r!(s-1)!}\sum_\sigma (-1)^\sigma\,\omega([X_{\sigma 1},\Phi(X_{\sigma 2},\ldots,X_{\sigma (r+1)})],X_{\sigma (r+2)},\ldots) \right.\\ &\left.-\frac{1}{(r-1)!(s-1)!2!}\sum_\sigma\,(-1)^\sigma\,\omega(\Phi([X_{\sigma 1},X_{\sigma 2}],X_{\sigma 3},\ldots),X_{\sigma (r+2)},\ldots)\right) \end{align*}

Again it is instructive to calculate the action of  i_{\Phi} and of this extended Lie derivative for \Phi=id_E. Using the local formula from [1] or [2]

 i_\Phi\omega=\sum\,\Phi^A_{B_1\ldots B_r}\,\omega_{AB_{r+1}\ldots B_{r+s}}\,dx^B

we obtain that for  \Phi=id_E, that is for \Phi^A_B=\delta^A_B, we have


But the identity map is not a graded derivative (of degree 0 in this case). On the other hand, from the formula involving vector fields I am getting

i_\phi\omega = s\omega

This looks good, because then

i_\Phi(\omega_1\wedge\omega_2)=(s_1\omega_1)\wedge\omega_2+\omega_1\wedge (s_2\omega_1)=(s_1+s_2)\omega_1\wedge\omega_2

as it should be.

At this point I have to pause until this point is clarified. Perhaps I am missing something evident? I don’t know.

And here are the conventions concerning differential forms used in [1] and [2]:

Differential Forms - Conventions in "Topics in Differential Geometry"


[1] Peter W. Michor, Topics in Differential Geometry (see hxxp:// )
[2] Ivan Kolar, Peter W. Michor, Jan Slovak, Natural Operations in Differentila Geometry (see hxxp:// )
[3] Andreas Kriegl, Peter W. Michor, The Convenient Setting for Global Analysis, Ch. 33.18