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In mathematics, the Frölicher–Nijenhuis bracket is an extension of the Lie bracket of vector fields to vector-valued differential forms on a differentiable manifold. It is useful in the study of connections, notably the Ehresmann connection, as well as in the more general study of projections in the tangent bundle. It was introduced by Alfred Frölicher and Albert Nijenhuis (1956) and is related to the work of Schouten (1940).

It is related to but not the same as the Nijenhuis–Richardson bracket and the Schouten–Nijenhuis bracket.

Definition

Let Ω*(M) be the sheaf of exterior algebras of differential forms on a smooth manifold M. This is a graded algebra in which forms are graded by degree:

${\mathrm{\Omega }}^{*}(M)=\underset{k=0}{\overset{\mathrm{\infty }}{⨁}}{\mathrm{\Omega }}^k(M).$

A graded derivation of degree ℓ is a mapping

$D:{\mathrm{\Omega }}^{*}(M)\to {\mathrm{\Omega }}^{*+l}(M)$

which is linear with respect to constants and satisfies

$D(\alpha \wedge \beta )=D(\alpha )\wedge \beta +(-1{)}^{\ell \mathrm{deg}(\alpha )}\alpha \wedge D(\beta ).$

Thus, in particular, the interior product with a vector defines a graded derivation of degree ℓ = −1, whereas the exterior derivative is a graded derivation of degree ℓ = 1.

The vector space of all derivations of degree ℓ is denoted by Der_ℓΩ*(M). The direct sum of these spaces is a graded vector space whose homogeneous components consist of all graded derivations of a given degree; it is denoted

$\mathrm{D}\mathrm{e}\mathrm{r}{\mathrm{\Omega }}^{*}(M)=\underset{k=-\mathrm{\infty }}{\overset{\mathrm{\infty }}{⨁}}{\mathrm{D}\mathrm{e}\mathrm{r}}_k{\mathrm{\Omega }}^{*}(M).$

This forms a graded Lie algebra under the anticommutator of derivations defined on homogeneous derivations D₁ and D₂ of degrees d₁ and d₂, respectively, by

$[D_1,D_2]=D_1\circ D_2-(-1{)}^{d_1{d}_2}{D}_2\circ D_1.$

Any vector-valued differential form K in Ω^k(M, TM) with values in the tangent bundle of M defines a graded derivation of degree k − 1, denote by i_K, and called the insertion operator. For ω ∈ Ω^ℓ(M),

$i_K\omega (X_1,\dots ,X_{k+\ell -1})=\frac{1}{k!(\ell -1)!}\sum _{\sigma \in S_{k+\ell -1}}\text{<mrow data-mjx-texclass="ORD">sign</mrow>}\sigma \cdot \omega (K(X_{\sigma (1)},\dots ,X_{\sigma (k)}),X_{\sigma (k+1)},\dots ,X_{\sigma (k+\ell -1)})$

The Nijenhuis–Lie derivative along K ∈ Ω^k(M, TM) is defined by

${\mathcal{ℒ}}_K=[d,i_K]=d\circ i_K-(-1{)}^{k-1}{i}_K\circ d$

where d is the exterior derivative and i_K is the insertion operator.

The Frölicher–Nijenhuis bracket is defined to be the unique vector-valued differential form

$[\cdot ,\cdot ]:{\mathrm{\Omega }}^k(M,\mathrm{T}M)\times {\mathrm{\Omega }}^{\ell }(M,\mathrm{T}M)\to {\mathrm{\Omega }}^{k+\ell }(M,\mathrm{T}M):(K,L)\mapsto [K,L]$

such that

${\mathcal{ℒ}}_{[K,L]}=[{\mathcal{ℒ}}_K,{\mathcal{ℒ}}_L].$

If k = 0, so that K ∈ Ω⁰(M, TM) is a vector field, the usual homotopy formula for the Lie derivative is recovered

${\mathcal{ℒ}}_K=[d,i_K]=d\circ i_K+i_K\circ d.$

An explicit formula for the Frölicher–Nijenhuis bracket of $\varphi \otimes X$ and $\psi \otimes Y$ (for forms φ and ψ and vector fields X and Y) is given by

$[\varphi \otimes X,\psi \otimes Y]=\varphi \wedge \psi \otimes [X,Y]+\varphi \wedge {\mathcal{ℒ}}_X\psi \otimes Y-{\mathcal{ℒ}}_Y\varphi \wedge \psi \otimes X+(-1{)}^{\mathrm{deg}(\varphi )}(d\varphi \wedge i_X(\psi )\otimes Y+i_Y(\varphi )\wedge d\psi \otimes X).$

Derivations of the ring of forms

Every derivation of Ω^*(M) can be written as

$i_L+{\mathcal{ℒ}}_K$

for unique elements K and L of Ω^*(M, TM). The Lie bracket of these derivations is given as follows.

• The derivations of the form ${\mathcal{ℒ}}_K$ form the Lie superalgebra of all derivations commuting with d. The bracket is given by

$[{\mathcal{ℒ}}_{K_1},{\mathcal{ℒ}}_{K_2}]={\mathcal{ℒ}}_{[K_1,K_2]}$

where the bracket on the right is the Frölicher–Nijenhuis bracket. In particular the Frölicher–Nijenhuis bracket defines a graded Lie algebra structure on $\mathrm{\Omega }(M,\mathrm{T}M)$, which extends the Lie bracket of vector fields.

• The derivations of the form $i_L$ form the Lie superalgebra of all derivations vanishing on functions Ω⁰(M). The bracket is given by

$[i_{L_1},i_{L_2}]=i_{[L_1,L_2{]}^{\wedge }}$

where the bracket on the right is the Nijenhuis–Richardson bracket.

• The bracket of derivations of different types is given by

$[{\mathcal{ℒ}}_K,i_L]=i_{[K,L]}-(-1{)}^{kl}{\mathcal{ℒ}}_{i_{L}K}$

for K in Ω^k(M, TM), L in Ω^l+1(M, TM).

Applications

The Nijenhuis tensor of an almost complex structure J, is the Frölicher–Nijenhuis bracket of J with itself. An almost complex structure is a complex structure if and only if the Nijenhuis tensor is zero.

With the Frölicher–Nijenhuis bracket it is possible to define the curvature and cocurvature of a vector-valued 1-form which is a projection. This generalizes the concept of the curvature of a connection.

There is a common generalization of the Schouten–Nijenhuis bracket and the Frölicher–Nijenhuis bracket; for details see the article on the Schouten–Nijenhuis bracket.

References

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  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
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  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
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