Pontryagin duality

In mathematics, a frame bundle is a principal fiber bundle F(E) associated to any vector bundle E. The fiber of F(E) over a point x is the set of all ordered bases, or frames, for E_x. The general linear group acts naturally on F(E) via a change of basis, giving the frame bundle the structure of a principal GL(k, R)-bundle (where k is the rank of E).

The frame bundle of a smooth manifold is the one associated to its tangent bundle. For this reason it is sometimes called the tangent frame bundle.

Definition and construction

Let E → X be a real vector bundle of rank k over a topological space X. A frame at a point x ∈ X is an ordered basis for the vector space E_x. Equivalently, a frame can be viewed as a linear isomorphism

$p:\mathbf {R} ^{k}\to E_{x}.$

The set of all frames at x, denoted F_x, has a natural right action by the general linear group GL(k, R) of invertible k × k matrices: a group element g ∈ GL(k, R) acts on the frame p via composition to give a new frame

$p\circ g:\mathbf {R} ^{k}\to E_{x}.$

This action of GL(k, R) on F_x is both free and transitive (This follows from the standard linear algebra result that there is a unique invertible linear transformation sending one basis onto another). As a topological space, F_x is homeomorphic to GL(k, R) although it lacks a group structure, since there is no "preferred frame". The space F_x is said to be a GL(k, R)-torsor.

The frame bundle of E, denoted by F(E) or F_GL(E), is the disjoint union of all the F_x:

$\mathrm {F} (E)=\coprod _{x\in X}F_{x}.$

Each point in F(E) is a pair (x, p) where x is a point in X and p is a frame at x. There is a natural projection π : F(E) → X which sends (x, p) to x. The group GL(k, R) acts on F(E) on the right as above. This action is clearly free and the orbits are just the fibers of π.

The frame bundle F(E) can be given a natural topology and bundle structure determined by that of E. Let (U_i, φ_i) be a local trivialization of E. Then for each x ∈ U_i one has a linear isomorphism φ_i,x : E_x → R^k. This data determines a bijection

$\psi _{i}:\pi ^{-1}(U_{i})\to U_{i}\times \mathrm {GL} (k,\mathbf {R} )$

given by

$\psi _{i}(x,p)=(x,\varphi _{i,x}\circ p).$

With these bijections, each π⁻¹(U_i) can be given the topology of U_i × GL(k, R). The topology on F(E) is the final topology coinduced by the inclusion maps π⁻¹(U_i) → F(E).

With all of the above data the frame bundle F(E) becomes a principal fiber bundle over X with structure group GL(k, R) and local trivializations ({U_i}, {ψ_i}). One can check that the transition functions of F(E) are the same as those of E.

The above all works in the smooth category as well: if E is a smooth vector bundle over a smooth manifold M then the frame bundle of E can be given the structure of a smooth principal bundle over M.

Associated vector bundles

A vector bundle E and its frame bundle F(E) are associated bundles. Each one determines the other. The frame bundle F(E) can be constructed from E as above, or more abstractly using the fiber bundle construction theorem. With the latter method, F(E) is the fiber bundle with same base, structure group, trivializing neighborhoods, and transition functions as E but with abstract fiber GL(k, R), where the action of structure group GL(k, R) on the fiber GL(k, R) is that of left multiplication.

Given any linear representation ρ : GL(k, R) → GL(V,F) there is a vector bundle

$\mathrm {F} (E)\times _{\rho }V$

associated to F(E) which is given by product F(E) × V modulo the equivalence relation (pg, v) ~ (p, ρ(g)v) for all g in GL(k, R). Denote the equivalence classes by [p, v].

The vector bundle E is naturally isomorphic to the bundle F(E) ×_ρ R^k where ρ is the fundamental representation of GL(k, R) on R^k. The isomorphism is given by

$[p,v]\mapsto p(v)$

where v is a vector in R^k and p : R^k → E_x is a frame at x. One can easily check that this map is well-defined.

Any vector bundle associated to E can be given by the above construction. For example, the dual bundle of E is given by F(E) ×_ρ* (R^k)* where ρ* is the dual of the fundamental representation. Tensor bundles of E can be constructed in a similar manner.

Tangent frame bundle

The tangent frame bundle (or simply the frame bundle) of a smooth manifold M is the frame bundle associated to the tangent bundle of M. The frame bundle of M is often denoted FM or GL(M) rather than F(TM). If M is n-dimensional then the tangent bundle has rank n, so the frame bundle of M is a principal GL(n, R) bundle over M.

Smooth frames

Local sections of the frame bundle of M are called smooth frames on M. The cross-section theorem for principal bundles states that the frame bundle is trivial over any open set in U in M which admits a smooth frame. Given a smooth frame s : U → FU, the trivialization ψ : FU → U × GL(n, R) is given by

$\psi (p)=(x,s(x)^{-1}\circ p)$

where p is a frame at x. It follows that a manifold is parallelizable if and only if the frame bundle of M admits a global section.

Since the tangent bundle of M is trivializable over coordinate neighborhoods of M so is the frame bundle. In fact, given any coordinate neighborhood U with coordinates (x¹,…,xⁿ) the coordinate vector fields

$\left({\frac {\partial }{\partial x^{1}}},\cdots ,{\frac {\partial }{\partial x^{n}}}\right)$

define a smooth frame on U. One of the advantages of working with frame bundles is that they allow one to work with frames other than coordinates frames; one can choose a frame adapted to the problem at hand. This is sometimes called the method of moving frames.

Solder form

The frame bundle of a manifold M is a special type of principal bundle in the sense that its geometry is fundamentally tied to the geometry of M. This relationship can be expressed by means of a vector-valued 1-form on FM called the solder form (also known as the fundamental or tautological 1-form). Let x be a point of the manifold M and p a frame at x, so that

$p:\mathbf {R} ^{n}\to T_{x}M$

is a linear isomorphism of Rⁿ with the tangent space of M at x. The solder form of FM is the Rⁿ-valued 1-form θ defined by

$\theta _{p}(\xi )=p^{-1}\mathrm {d} \pi (\xi )$

where ξ is a tangent vector to FM at the point (x,p), p⁻¹ : T_xM → Rⁿ is the inverse of the frame map, and dπ is the differential of the projection map π : FM → M. The solder form is horizontal in the sense that it vanishes on vectors tangent to the fibers of π and right equivariant in the sense that

$R_{g}^{*}\theta =g^{-1}\theta$

where R_g is right translation by g ∈ GL(n, R). A form with these properties is called a basic or tensorial form on FM. Such forms are in 1-1 correspondence with TM-valued 1-forms on M which are, in turn, in 1-1 correspondence with smooth bundle maps TM → TM over M. Viewed in this light θ is just the identity map on TM.

Orthonormal frame bundle

If a vector bundle E is equipped with a Riemannian bundle metric then each fiber E_x is not only a vector space but an inner product space. It is then possible to talk about the set of all of orthonormal frames for E_x. An orthonormal frame for E_x is an ordered orthonormal basis for E_x, or, equivalently, a linear isometry

$p:\mathbf {R} ^{k}\to E_{x}$

where R^k is equipped with the standard Euclidean metric. The orthogonal group O(k) acts freely and transitively on the set of all orthonormal frames via right composition. In other words, the set of all orthonormal frames is a right O(k)-torsor.

The orthonormal frame bundle of E, denoted F_O(E), is the set of all orthonormal frames at each point x in the base space X. It can be constructed by a method entirely analogous to that of the ordinary frame bundle. The orthonormal frame bundle of a rank k Riemannian vector bundle E → X is a principal O(k)-bundle over X. Again, the construction works just as well in the smooth category.

If the vector bundle E is orientable then one can define the oriented orthonormal frame bundle of E, denoted F_SO(E), as the principal SO(k)-bundle of all positively-oriented orthonormal frames.

If M is an n-dimensional Riemannian manifold, then the orthonormal frame bundle of M, denoted F_OM or O(M), is the orthonormal frame bundle associated to the tangent bundle of M (which is equipped with a Riemannian metric by definition). If M is orientable, then one also has the oriented orthonormal frame bundle F_SOM.

Given a Riemannian vector bundle E, the orthonormal frame bundle is a principal O(k)-subbundle of the general linear frame bundle. In other words, the inclusion map

$i:{\mathrm {F} }_{\mathrm {O} }(E)\to {\mathrm {F} }_{\mathrm {GL} }(E)$

is principal bundle map. One says that F_O(E) is a reduction of the structure group of F_GL(E) from GL(k, R) to O(k).

G-structures

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If a smooth manifold M comes with additional structure it is often natural to consider a subbundle of the full frame bundle of M which is adapted to the given structure. For example, if M is a Riemannian manifold we saw above that it is natural to consider the orthonormal frame bundle of M. The orthonormal frame bundle is just a reduction of the structure group of F_GL(M) to the orthogonal group O(n).

In general, if M is a smooth n-manifold and G is a Lie subgroup of GL(n, R) we define a G-structure on M to be a reduction of the structure group of F_GL(M) to G. Explicitly, this is a principal G-bundle F_G(M) over M together with a G-equivariant bundle map

${\mathrm {F} }_{G}(M)\to {\mathrm {F} }_{\mathrm {GL} }(M)$

over M.

In this language, a Riemannian metric on M gives rise to an O(n)-structure on M. The following are some other examples.

• Every oriented manifold has an oriented frame bundle which is just a GL⁺(n, R)-structure on M.
• A volume form on M determines a SL(n, R)-structure on M.
• A 2n-dimensional symplectic manifold has a natural Sp(2n, R)-structure.
• A 2n-dimensional complex or almost complex manifold has a natural GL(n, C)-structure.

In many of these instances, a G-structure on M uniquely determines the corresponding structure on M. For example, a SL(n, R)-structure on M determines a volume form on M. However, in some cases, such as for symplectic and complex manifolds, an added integrability condition is needed. A Sp(2n, R)-structure on M uniquely determines a nondegenerate 2-form on M, but for M to be symplectic, this 2-form must also be closed.

References

• 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.
  
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  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