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In [[mathematics]], the '''Chern–Weil homomorphism''' is a basic construction in the [[Chern–Weil theory]], relating for a [[smooth manifold]] ''M'' the [[curvature]] of ''M'' to the [[de Rham cohomology]] groups of ''M'', i.e., geometry to topology. This theory of [[Shiing-Shen Chern]] and [[André Weil]] from the 1940s was an important step in the theory of [[characteristic class]]es. It is a generalization of the [[Chern–Gauss–Bonnet theorem]]. | |||
Denote by <math>\mathbb K</math> either the [[real number|real field]] or [[complex field]]. Let ''G'' be a real or complex [[Lie group]] with [[Lie algebra]] <math>\mathfrak g</math>; and let | |||
:<math>\mathbb K(\mathfrak g^*)</math> | |||
denote the algebra of <math>\mathbb K</math>-valued [[Polynomials on vector spaces|polynomial]]s on <math>\mathfrak g</math>. Let <math>\mathbb K(\mathfrak g^*)^{Ad(G)}</math> be the subalgebra of fixed points in <math> \mathbb K(\mathfrak g^*)</math> under the [[adjoint representation of a Lie group|adjoint action]] of ''G'', so that for instance | |||
= | :<math>f(t_1,\dots,t_k)=f(Ad_g t_1,\dots, Ad_g t_k) \, </math> | ||
< | for all <math>f\in\mathbb K(\mathfrak g^*)^{Ad(G)}. \, </math> | ||
< | The '''Chern–Weil homomorphism''' is a homomorphism of <math>\mathbb K</math>-algebras from <math>\mathbb K(\mathfrak g^*)^{Ad(G)}</math> to the cohomology algebra <math>H^*(M,\mathbb K)</math>. Such a homomorphism exists and is uniquely defined for every [[principal bundle|principal G-bundle]] ''P'' on ''M''. If ''G'' is compact, then under the homomorphism, the cohomology ring of the [[classifying space]] for ''G''-bundles ''B''<sup>''G''</sup> is isomorphic to the algebra <math>\mathbb K(\mathfrak g^*)^{Ad(G)}</math> of invariant polynomials: | ||
:<math>H^*(B^G, \mathbb{K}) \cong \mathbb K(\mathfrak g^*)^{Ad(G)}.</math> | |||
For non-compact groups like SL(''n'','''R'''), there may be cohomology classes that are not represented by invariant polynomials. | |||
< | ==Definition of the homomorphism== | ||
Choose any [[connection form]] w in P, and let <math>\Omega</math> be the associated [[curvature]] 2-form. If <math>f\in\mathbb K(\mathfrak g^*)^{Ad(G)}</math> is a homogeneous polynomial of degree ''k'', let <math>f(\Omega)</math> be the 2''k''-form on ''P'' given by | |||
</ | :<math>f(\Omega)(X_1,\dots,X_{2k})=\frac{1}{(2k)!}\sum_{\sigma\in\mathfrak S_{2k}}\epsilon_\sigma f(\Omega(X_{\sigma(1)},X_{\sigma(2)}),\dots,\Omega(X_{\sigma(2k-1)}, X_{\sigma(2k)}))</math> | ||
where <math>\epsilon_\sigma</math> is the sign of the permutation <math>\sigma</math> in the symmetric group on 2''k'' numbers <math>\mathfrak S_{2k}</math>. | |||
(see [[Pfaffian]]). | |||
One can then show that | |||
< | :<math>f(\Omega)</math> | ||
is a [[closed differential form|closed form]], so that | |||
:<math>df(\Omega)=0, \, </math> | |||
and that the [[de Rham cohomology]] class of | |||
< | :<math>f(\Omega) \, </math> | ||
is independent of the choice of connection on ''P'', so it depends only upon the principal bundle. | |||
Thus letting | |||
:<math>\phi(f) \, </math> | |||
be the cohomology class obtained in this way from f, we obtain an algebra homomorphism | |||
:<math>\phi:\mathbb K(\mathfrak g^*)^{Ad(G)}\rightarrow H^*(M,\mathbb K). \, </math> | |||
==References== | |||
* {{citation|last=Bott|first=R.|authorlink=Raoul Bott|title=On the Chern–Weil homomorphism and the continuous cohomology of Lie groups|journal=Advances in Math| year=1973|volume=11|pages=289–303|doi=10.1016/0001-8708(73)90012-1}}. | |||
* {{citation|last=Chern|first=S.-S.|authorlink=S.-S. Chern|title=Topics in Differential Geometry|publisher=Institute for Advanced Study, mimeographed lecture notes|year=1951}}. | |||
* Shiing-Shen Chern, ''Complex Manifolds Without Potential Theory'' (Springer-Verlag Press, 1995) ISBN 0-387-90422-0, ISBN 3-540-90422-0. | |||
*:The appendix of this book: "Geometry of Characteristic Classes" is a very neat and profound introduction to the development of the ideas of characteristic classes. | |||
* {{citation|last1=Chern|first1=S.-S.|authorlink1=S.-S. Chern|last2=Simons|first2=J|authorlink2=James Harris Simons|title=Characteristic forms and geometric invariants|journal=The Annals of Mathematics. Second Series|volume=99|number=1|year=1974|pages=48–69|jstor=1971013}}. | |||
* {{citation|last1=Kobayashi|first1=S.|last2=Nomizu|first2=K.|title=[[Foundations of Differential Geometry]], Vol. 2|publisher=Wiley-Interscience|year=1963|publication-date= new ed. 2004}}. | |||
* {{citation|last1=Narasimhan|first1=M.|last2=Ramanan|first2=S.|title=Existence of universal connections|journal=Amer. J. Math.|volume=83|year=1961|pages=563–572|jstor=2372896|doi=10.2307/2372896}}. | |||
* {{citation|last1=Morita|first1=Shigeyuki|title=Geometry of Differential Forms|journal=A.M.S monograph|volume=201|year=2000}}. | |||
{{DEFAULTSORT:Chern-Weil homomorphism}} | |||
[[Category:Differential geometry]] | |||
[[Category:Characteristic classes]] | |||
Revision as of 12:53, 9 January 2014
In mathematics, the Chern–Weil homomorphism is a basic construction in the Chern–Weil theory, relating for a smooth manifold M the curvature of M to the de Rham cohomology groups of M, i.e., geometry to topology. This theory of Shiing-Shen Chern and André Weil from the 1940s was an important step in the theory of characteristic classes. It is a generalization of the Chern–Gauss–Bonnet theorem.
Denote by either the real field or complex field. Let G be a real or complex Lie group with Lie algebra ; and let
denote the algebra of -valued polynomials on . Let be the subalgebra of fixed points in under the adjoint action of G, so that for instance
for all
The Chern–Weil homomorphism is a homomorphism of -algebras from to the cohomology algebra . Such a homomorphism exists and is uniquely defined for every principal G-bundle P on M. If G is compact, then under the homomorphism, the cohomology ring of the classifying space for G-bundles BG is isomorphic to the algebra of invariant polynomials:
For non-compact groups like SL(n,R), there may be cohomology classes that are not represented by invariant polynomials.
Definition of the homomorphism
Choose any connection form w in P, and let be the associated curvature 2-form. If is a homogeneous polynomial of degree k, let be the 2k-form on P given by
where is the sign of the permutation in the symmetric group on 2k numbers .
(see Pfaffian).
One can then show that
is a closed form, so that
and that the de Rham cohomology class of
is independent of the choice of connection on P, so it depends only upon the principal bundle.
Thus letting
be the cohomology class obtained in this way from f, we obtain an algebra homomorphism
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.
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. - Shiing-Shen Chern, Complex Manifolds Without Potential Theory (Springer-Verlag Press, 1995) ISBN 0-387-90422-0, ISBN 3-540-90422-0.
- The appendix of this book: "Geometry of Characteristic Classes" is a very neat and profound introduction to the development of the ideas of characteristic classes.
- Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
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.
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. - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
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.
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. - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
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.
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. - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
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.
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.