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In differential geometry, a fibered manifold is surjective submersion of smooth manifolds $Y\to X$. Locally trivial fibered manifolds are fiber bundles. Therefore, a notion of connection on fibered manifolds provides a general framework of a connection on fiber bundles.

Formal definition

Let $\pi :Y\to X$ be a fibered manifold. A (generalized) connection on $Y$ is a section $\Gamma :Y\to J^{1}Y$, where $J^{1}Y$ is the jet manifold of $Y$.

Connection as a horizontal splitting

Let $\pi :Y\to X$ be a fibered manifold. There is the following canonical short exact sequence of vector bundles over $Y$:

$0\to VY\to TY\to Y\times _{X}TX\to 0,\qquad \qquad (1)$

where $TY$ and $TX$ are the tangent bundles of $Y$, respectively, $VY$ is the vertical tangent bundle of $Y$, and $Y\times _{X}TX$ is the pullback bundle of $TX$ onto $Y$.

A connection on a fibered manifold $Y\to X$ is defined as a linear bundle morphism

$\Gamma :Y\times _{X}TX\to TY\qquad \qquad (2)$

over $Y$ which splits the exact sequence (1). A connection always exists.

Sometimes, this connection $\Gamma$ is called the Ehresmann connection because it yields the horizontal distribution

$HY=\Gamma (Y\times _{X}TX)\subset TY$

of $TY$ and its horizontal decomposition $TY=VY\oplus HY$.

At the same time, by an Ehresmann connection also is meant the following construction. Any connection $\Gamma$ on a fibered manifold $Y\to X$ yields a horizontal lift $\Gamma \circ \tau$ of a vector field $\tau$ on $X$ onto $Y$, but need not defines the similar lift of a path in $X$ into $Y$. Let $\mathbb {R} \supset [,]\ni t\to x(t)\in X$ and $\mathbb {R} \ni t\to y(t)\in Y$ be smooth paths in $X$ and $Y$, respectively. Then $t\to y(t)$ is called the horizontal lift of $x(t)$ if $\pi (y(t))=x(t)$, ${\dot {y}}(t)\in HY$, $t\in \mathbb {R}$. A connection $\Gamma$ is said to be the Ehresmann connection if, for each path $x([0,1])$ in $X$, there exists its horizontal lift trough any point $y\in \pi ^{-1}(x([0,1]))$. A fibered manifold is a fiber bundle if and only if it admits such an Ehresmann connection.

Connection as a tangent-valued form

Given a fibered manifold $Y\to X$, let it be endowed with an atlas of fibered coordinates $(x^{\mu },y^{i})$, and let $\Gamma$ be a connection on $Y\to X$. It yields uniquely the horizontal tangent-valued one-form

$\Gamma =dx^{\lambda }\otimes (\partial _{\lambda }+\Gamma _{\lambda }^{i}(x^{\nu },y^{j})\partial _{i})\qquad \qquad (3)$

on $Y$ which projects onto the canonical tangent-valued form (tautological one-form) $\theta _{X}=dx^{\mu }\otimes \partial _{\mu }$ on $X$, and vice versa. With this form, the horizontal splitting (2) reads

$\Gamma :\partial _{\lambda }\to \partial _{\lambda }\rfloor \Gamma =\partial _{\lambda }+\Gamma _{\lambda }^{i}\partial _{i}.$

In particular, the connection $\Gamma$ (3) yields the horizontal lift of any vector field $\tau =\tau ^{\mu }\partial _{\mu }$ on $X$ to a projectable vector field

$\Gamma \tau =\tau \rfloor \Gamma =\tau ^{\lambda }(\partial _{\lambda }+\Gamma _{\lambda }^{i}\partial _{i})\subset HY$

on $Y$.

Connection as a vertical-valued form

The horizontal splitting (2) of the exact sequence (1) defines the corresponding splitting of the dual exact sequence

$0\to Y\times _{X}T^{*}X\to T^{*}Y\to V^{*}Y\to 0,$

where $T^{*}Y$ and $T^{*}X$ are the cotangent bundles of $Y$, respectively, and $V^{*}Y\to Y$ is the dual bundle to $VY\to Y$, called the vertical cotangent bundle. This splitting is given by the vertical-valued form

$\Gamma =(dy^{i}-\Gamma _{\lambda }^{i}dx^{\lambda })\otimes \partial _{i},$

which also represents a connection on a fibered manifold.

Treating a connection as a vertical-valued form, one comes to the following important construction. Given a fibered manifold $Y\to X$, let $f:X'\to X$ be a morphism and $f^{*}Y\to X'$ the pullback bundle of $Y$ by $f$. Then any connection $\Gamma$ (3) on $Y\to X$ induces the pullback connection

$f^{*}\Gamma =(dy^{i}-(\Gamma \circ {\widetilde {f}})_{\lambda }^{i}{\frac {\partial f^{\lambda }}{\partial x'^{\mu }}}dx'^{\mu })\otimes \partial _{i}$

on $f^{*}Y\to X'$.

Connection as a jet bundle section

Let $J^{1}Y$ be the jet manifold of sections of a fibered manifold $Y\to X$, with coordinates $(x^{\mu },y^{i},y_{\mu }^{i})$. Due to the canonical imbedding

$J^{1}Y\to _{Y}(Y\times _{X}T^{*}X)\otimes _{Y}TY,\qquad (y_{\mu }^{i})\to dx^{\mu }\otimes (\partial _{\mu }+y_{\mu }^{i}\partial _{i}),$

any connection $\Gamma$ (3) on a fibered manifold $Y\to X$ is represented by a global section

$\Gamma :Y\to J^{1}Y,\qquad y_{\lambda }^{i}\circ \Gamma =\Gamma _{\lambda }^{i},$

of the jet bundle $J^{1}Y\to Y$, and vice versa. It is an affine bundle modelled on a vector bundle

$(Y\times _{X}T^{*}X)\otimes _{Y}VY\to Y.\qquad \qquad (4)$

There are the following corollaries of this fact.

(i) Connections on a fibered manifold $Y\to X$ make up an affine space modelled on the vector space of soldering forms

$\sigma =\sigma _{\mu }^{i}dx^{\mu }\otimes \partial _{i}\qquad \qquad (5)$

on $Y\to X$, i.e., sections of the vector bundle (4).

(ii) Connection coefficients possess the coordinate transformation law

${\Gamma '}_{\lambda }^{i}={\frac {\partial x^{\mu }}{\partial {x'}^{\lambda }}}(\partial _{\mu }{y'}^{i}+\Gamma _{\mu }^{j}\partial _{j}{y'}^{i}).$

(iii) Every connection $\Gamma$ on a fibred manifold $Y\to X$ yields the first order differential operator

$D_{\Gamma }:J^{1}Y\to _{Y}T^{*}X\otimes _{Y}VY,\qquad D_{\Gamma }=(y_{\lambda }^{i}-\Gamma _{\lambda }^{i})dx^{\lambda }\otimes \partial _{i},$

on $Y$ called the covariant differential relative to the connection $\Gamma$. If $s:X\to Y$ is a section, its covariant differential

$\nabla ^{\Gamma }s=(\partial _{\lambda }s^{i}-\Gamma _{\lambda }^{i}\circ s)dx^{\lambda }\otimes \partial _{i},$

and the covariant derivative $\nabla _{\tau }^{\Gamma }s=\tau \rfloor \nabla ^{\Gamma }s$ along a vector field $\tau$ on $X$ are defined.

Curvature and torsion

Given the connection $\Gamma$ (3) on a fibered manifold $Y\to X$, its curvature is defined as the Nijenhuis differential

$R={\frac {1}{2}}d_{\Gamma }\Gamma ={\frac {1}{2}}[\Gamma ,\Gamma ]_{FN}={\frac {1}{2}}R_{\lambda \mu }^{i}\,dx^{\lambda }\wedge dx^{\mu }\otimes \partial _{i},$

$R_{\lambda \mu }^{i}=\partial _{\lambda }\Gamma _{\mu }^{i}-\partial _{\mu }\Gamma _{\lambda }^{i}+\Gamma _{\lambda }^{j}\partial _{j}\Gamma _{\mu }^{i}-\Gamma _{\mu }^{j}\partial _{j}\Gamma _{\lambda }^{i}.$

This is a vertical-valued horizontal two-form on $Y$.

Given the connection $\Gamma$ (3) and the soldering form $\sigma$ (5), a torsion of $\Gamma$ with respect to $\sigma$ is defined as

$T=d_{\Gamma }\sigma =(\partial _{\lambda }\sigma _{\mu }^{i}+\Gamma _{\lambda }^{j}\partial _{j}\sigma _{\mu }^{i}-\partial _{j}\Gamma _{\lambda }^{i}\sigma _{\mu }^{j})\,dx^{\lambda }\wedge dx^{\mu }\otimes \partial _{i}.$

Bundle of principal connections

Let $\pi \colon P\to M$ be a principal bundle with a structure Lie group $G$. A principal connection on $P$ usually is described by a Lie algebra-valued connection one-form on $P$. At the same time, a principal connection on $P$ is a global section of the jet bundle $J^{1}P\to P$ which is equivariant with respect to the canonical right action of $G$ in $P$. Therefore, it is represented by a global section of the quotient bundle $C=J^{1}P/G\to M$, called the bundle of principal connections. It is an affine bundle modelled on the vector bundle $VP/G\to M$ whose typical fiber is the Lie algebra ${\mathfrak {g}}$ of structure group $G$, and where $G$ acts on by the adjoint representation. There is the canonical imbedding of $C$ to the quotient bundle $TP/G$ which also is called the bundle of principal connections.

Given a basis $\{{\mathrm {e} }_{m}\}$ for a Lie algebra of $G$, the fiber bundle $C$ is endowed with bundle coordinates $(x^{\mu },a_{\mu }^{m})$, and its sections are represented by vector-valued one-forms

$A=dx^{\lambda }\otimes (\partial _{\lambda }+a_{\lambda }^{m}{\mathrm {e} }_{m}),$

where $a_{\lambda }^{m}\,dx^{\lambda }\otimes {\mathrm {e} }_{m}$ are the familiar local connection forms on $M$.

Let us note that the jet bundle $J^{1}C$ of $C$ is a configuration space of Yang–Mills gauge theory. It admits the canonical decomposition

$a_{\lambda \mu }^{r}={\frac {1}{2}}(F_{\lambda \mu }^{r}+S_{\lambda \mu }^{r})={\frac {1}{2}}(a_{\lambda \mu }^{r}+a_{\mu \lambda }^{r}-c_{pq}^{r}a_{\lambda }^{p}a_{\mu }^{q})+{\frac {1}{2}}(a_{\lambda \mu }^{r}-a_{\mu \lambda }^{r}+c_{pq}^{r}a_{\lambda }^{p}a_{\mu }^{q}),$

where

$F={\frac {1}{2}}F_{\lambda \mu }^{m}\,dx^{\lambda }\wedge dx^{\mu }\otimes {\mathrm {e} }_{m}$

is called the strength form of a principal connection.

See also

• Connection (mathematics)
• Fibred manifold
• Ehresmann connection
• Connection (principal bundle)

Notes

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References

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  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
• Mangiarotti, L., Sardanashvily, G., Connections in Classical and Quantum Field Theory. World Scientific, 2000. ISBN 981-02-2013-8.

External links

• Sardanashvily, G., Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory, Lambert Academic Publishing, 2013. ISBN 978-3-659-37815-7; arXiv: 0908.1886