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In [[mathematics]], the '''Frölicher–Nijenhuis bracket''' is an extension of the [[Lie bracket of vector fields|Lie bracket]] of [[vector fields]] to [[vector-valued differential form]]s on a [[differentiable manifold]]. It is useful in the study of [[connection (mathematics)|connections]], notably the [[Ehresmann connection]], as well as in the more general study of [[projection (linear algebra)|projections]] in the [[tangent bundle]]. | |||
It was introduced by [[Alfred Frölicher]] and [[Albert Nijenhuis]] (1956) and is related to the work of [[Jan Arnoldus Schouten|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 (mathematics)|sheaf]] of [[exterior algebra]]s of [[differential form]]s on a [[smooth manifold]] ''M''. This is a [[graded algebra]] in which forms are graded by degree: | |||
:<math>\Omega^*(M) = \bigoplus_{k=0}^\infty \Omega^k(M).</math> | |||
A [[graded derivation]] of degree ℓ is a mapping | |||
:<math>D:\Omega^*(M)\to\Omega^{*+l}(M)</math> | |||
which is linear with respect to constants and satisfies | |||
:<math>D(\alpha\wedge\beta) = D(\alpha)\wedge\beta + (-1)^{\ell\deg(\alpha)}\alpha\wedge D(\beta).</math> | |||
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<sub>ℓ</sub>Ω*(''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 | |||
:<math>\mathrm{Der}\, \Omega^*(M) = \bigoplus_{k=-\infty}^\infty \mathrm{Der}_k\, \Omega^*(M).</math> | |||
This forms a [[graded Lie algebra]] under the anticommutator of derivations defined on homogeneous derivations ''D''<sub>1</sub> and ''D''<sub>2</sub> of degrees ''d''<sub>1</sub> and ''d''<sub>2</sub>, respectively, by | |||
:<math>[D_1,D_2] = D_1\circ D_2 - (-1)^{d_1d_2}D_2\circ D_1.</math> | |||
Any [[vector-valued differential form]] ''K'' in Ω<sup>''k''</sup>(''M'', T''M'') with values in the [[tangent bundle]] of ''M'' defines a graded derivation of degree ''k'' − 1, denote by ''i''<sub>''K''</sub>, and called the insertion operator. For ω ∈ Ω<sup>ℓ</sup>(''M''), | |||
:<math>i_K\,\omega(X_1,\dots,X_{k+\ell-1})=\frac{1}{k!(\ell-1)!}\sum_{\sigma\in{S}_{k+\ell-1}}\textrm{sign}\,\sigma \cdot | |||
\omega(K(X_{\sigma(1)},\dots,X_{\sigma(k)}),X_{\sigma(k+1)},\dots,X_{\sigma(k+\ell-1)}) | |||
</math> | |||
The [[Lie derivative#Nijenhuis–Lie derivative|Nijenhuis–Lie derivative]] along ''K'' ∈ Ω<sup>k</sup>(''M'', T''M'') is defined by | |||
:<math>\mathcal{L}_K = [d,i_K] =d\,{\circ}\, i_K-(-1)^{k-1}i_K{\circ}\, d</math> | |||
where ''d'' is the exterior derivative and ''i''<sub>K</sub> is the insertion operator. | |||
The Frölicher–Nijenhuis bracket is defined to be the unique vector-valued differential form | |||
:<math>[\cdot, \cdot] : \Omega^k(M,\mathrm{T}M) \times \Omega^\ell(M,\mathrm{T}M) \to \Omega^{k+\ell}(M,\mathrm{T}M) : (K, L) \mapsto [K, L]</math> | |||
such that | |||
:<math>\mathcal{L}_{[K, L]} = [\mathcal{L}_K, \mathcal{L}_L].</math> | |||
If ''k'' = 0, so that ''K'' ∈ Ω<sup>0</sup>(''M'', T''M'') | |||
is a vector field, the usual homotopy formula for the Lie derivative is recovered | |||
:<math>\mathcal{L}_K = [d,i_K] =d \,{\circ}\, i_K+i_K \,{\circ}\, d.</math> | |||
An explicit formula for the Frölicher–Nijenhuis bracket of <math>\phi\otimes X</math> and <math>\psi\otimes Y</math> (for forms φ and ψ and vector fields ''X'' and ''Y'') is given by | |||
:<math>\left.\right.[\phi \otimes X,\psi \otimes Y] = \phi\wedge\psi\otimes [X,Y] + \phi\wedge\mathcal{L}_X \psi\otimes Y - \mathcal{L}_Y \phi\wedge\psi \otimes X +(-1)^{\deg(\phi)}(d\phi \wedge i_X(\psi)\otimes Y +i_Y(\phi) \wedge d\psi \otimes X).</math> | |||
==Derivations of the ring of forms== | |||
Every derivation of Ω<sup>*</sup>(''M'') can be written as | |||
:<math>i_L + \mathcal{L}_K</math> | |||
for unique elements ''K'' and ''L'' of Ω<sup>*</sup>(''M'', T''M''). The Lie bracket of these derivations is given as follows. | |||
*The derivations of the form <math>\mathcal{L}_K</math> form the Lie superalgebra of all derivations commuting with ''d''. The bracket is given by | |||
::<math>[\mathcal{L}_{K_1},\mathcal{L}_{K_2}]= \mathcal{L}_{[K_1,K_2]}</math> | |||
:where the bracket on the right is the Frölicher–Nijenhuis bracket. In particular the Frölicher–Nijenhuis bracket defines a [[graded Lie algebra]] structure on <math>\Omega(M,\mathrm{T}M)</math>, which extends the [[Lie bracket of vector fields|Lie bracket]] of [[vector field]]s. | |||
*The derivations of the form <math>i_L</math> form the Lie superalgebra of all derivations vanishing on functions Ω<sup>0</sup>(''M''). The bracket is given by | |||
::<math>[i_{L_1},i_{L_2}]= i_{[L_1,L_2]^\and}</math> | |||
:where the bracket on the right is the [[Nijenhuis–Richardson bracket]]. | |||
*The bracket of derivations of different types is given by | |||
::<math>[\mathcal{L}_{K}, i_L]= i_{[K,L]} - (-1)^{kl}\mathcal{L}_{i_LK}</math> | |||
: for ''K'' in Ω<sup>k</sup>(''M'', T''M''), ''L'' in Ω<sup>l+1</sup>(''M'', T''M''). | |||
==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 (mathematics)|projection]]. This generalizes the concept of the curvature of a [[Connection (mathematics)|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== | |||
*{{citation | |||
| last1 = Frölicher | first1 = A. | |||
| last2 = Nijenhuis | first2 = A. | author2-link = Albert Nijenhuis | |||
| journal = [[Indagationes Mathematicae]] | |||
| pages = 338–360 | |||
| title = Theory of vector valued differential forms. Part I. | |||
| volume = 18 | |||
| year = 1956}}. | |||
*{{citation | |||
| last1 = Frölicher | first1 = A. | |||
| last2 = Nijenhuis | first2 = A. | author2-link = Albert Nijenhuis | |||
| journal = Communicationes Mathematicae Helveticae | |||
| pages = 227–248 | |||
| title = Invariance of vector form operations under mappings | |||
| volume = 34 | |||
| year = 1960}}. | |||
*{{springer|id=F/f120230|author=P. W. Michor|title=Frölicher–Nijenhuis bracket}} | |||
*{{citation | |||
| last = Schouten | first = J. A. | author-link = Jan Arnoldus Schouten | |||
| journal = [[Indagationes Mathematicae]] | |||
| pages = 449–452 | |||
| title = Über Differentialkonkomitanten zweier kontravarianten Grössen | |||
| volume = 2 | |||
| year = 1940}}. | |||
{{DEFAULTSORT:Frolicher-Nijenhuis Bracket}} | |||
[[Category:Bilinear operators]] | |||
[[Category:Binary operations]] | |||
[[Category:Differential geometry]] | |||
Revision as of 04:10, 27 January 2014
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:
A graded derivation of degree ℓ is a mapping
which is linear with respect to constants and satisfies
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 DerℓΩ*(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
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
![{\displaystyle [D_{1},D_{2}]=D_{1}\circ D_{2}-(-1)^{d_{1}d_{2}}D_{2}\circ D_{1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1adfbb0ec04a749db93ad72a82fb262aa19645f4)
Any vector-valued differential form K in Ωk(M, TM) with values in the tangent bundle of M defines a graded derivation of degree k − 1, denote by iK, and called the insertion operator. For ω ∈ Ωℓ(M),
The Nijenhuis–Lie derivative along K ∈ Ωk(M, TM) is defined by
![{\displaystyle {\mathcal {L}}_{K}=[d,i_{K}]=d\,{\circ }\,i_{K}-(-1)^{k-1}i_{K}{\circ }\,d}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34b2146b6e3c77b8f999a0ae507c13d4f50099ae)
where d is the exterior derivative and iK is the insertion operator.
The Frölicher–Nijenhuis bracket is defined to be the unique vector-valued differential form
![[\cdot ,\cdot ]:\Omega ^{k}(M,{\mathrm {T}}M)\times \Omega ^{\ell }(M,{\mathrm {T}}M)\to \Omega ^{{k+\ell }}(M,{\mathrm {T}}M):(K,L)\mapsto [K,L]](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8189832e245e1d375865a2f4758475c98b21185)
such that
![{\displaystyle {\mathcal {L}}_{[K,L]}=[{\mathcal {L}}_{K},{\mathcal {L}}_{L}].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c53e153859947365b390dd098bc09b3089118fe9)
If k = 0, so that K ∈ Ω0(M, TM) is a vector field, the usual homotopy formula for the Lie derivative is recovered
![{\displaystyle {\mathcal {L}}_{K}=[d,i_{K}]=d\,{\circ }\,i_{K}+i_{K}\,{\circ }\,d.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c38022109def3c76150428c57bcf26a7ab5b3b63)
An explicit formula for the Frölicher–Nijenhuis bracket of and (for forms φ and ψ and vector fields X and Y) is given by
![\left.\right.[\phi \otimes X,\psi \otimes Y]=\phi \wedge \psi \otimes [X,Y]+\phi \wedge {\mathcal {L}}_{X}\psi \otimes Y-{\mathcal {L}}_{Y}\phi \wedge \psi \otimes X+(-1)^{{\deg(\phi )}}(d\phi \wedge i_{X}(\psi )\otimes Y+i_{Y}(\phi )\wedge d\psi \otimes X).](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bdc84f70e846b15717d988516d45d7a3e746757)
Derivations of the ring of forms
Every derivation of Ω*(M) can be written as
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 form the Lie superalgebra of all derivations commuting with d. The bracket is given by
- where the bracket on the right is the Frölicher–Nijenhuis bracket. In particular the Frölicher–Nijenhuis bracket defines a graded Lie algebra structure on , which extends the Lie bracket of vector fields.
![[{\mathcal {L}}_{{K_{1}}},{\mathcal {L}}_{{K_{2}}}]={\mathcal {L}}_{{[K_{1},K_{2}]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c25a3a5670af1e2cdcda1b22a19fd47d059a13dd)
- The derivations of the form form the Lie superalgebra of all derivations vanishing on functions Ω0(M). The bracket is given by
- where the bracket on the right is the Nijenhuis–Richardson bracket.
![{\displaystyle [i_{L_{1}},i_{L_{2}}]=i_{[L_{1},L_{2}]^{\land }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2094b04589ac2f52cfe65b940a4886b7549e3ea4)
- The bracket of derivations of different types is given by
- for K in Ωk(M, TM), L in Ωl+1(M, TM).
![{\displaystyle [{\mathcal {L}}_{K},i_{L}]=i_{[K,L]}-(-1)^{kl}{\mathcal {L}}_{i_{L}K}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4664e2c2a82ba60bb004d1a6055e374b0c2e057c)
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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In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
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In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
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.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.