connection (fibred manifold)

Let {{math| π : Y → X}} be a fibered manifold. A generalized connection on {{mvar|Y}} is a section {{math|Γ : Y → J¹Y}}, where {{math|J¹Y}} is the jet manifold of {{mvar|Y}}.{{cite book|last1 = Krupka|first1=Demeter|last2=Janyška|first2=Josef|title=Lectures on differential invariants|year = 1990|publisher = Univerzita J. E. Purkyně v Brně|isbn=80-210-0165-8|page=174}}

With the above manifold {{mvar|π}} there is the following canonical short exact sequence of vector bundles over {{mvar|Y}}:

{{NumBlk|:|$0\backslash to\; \backslash mathrm\{V\}Y\backslash to\; \backslash mathrm\{T\}Y\backslash to\; Y\backslash times\_X\; \backslash mathrm\{T\}X\backslash to\; 0\backslash ,,$|{{EquationRef|1}}}}

where {{math|TY}} and {{math|TX}} are the tangent bundles of {{mvar|Y}}, respectively, {{math|VY}} is the vertical tangent bundle of {{mvar|Y}}, and {{math|Y ×_X TX}} is the pullback bundle of {{math|TX}} onto {{mvar|Y}}.

A connection on a fibered manifold {{math|Y → X}} is defined as a linear bundle morphism

{{NumBlk|:|$\backslash Gamma:\; Y\backslash times\_X\; \backslash mathrm\{T\}X\; \backslash to\; \backslash mathrm\{T\}Y$|{{EquationRef|2}}}}

over {{mvar|Y}} which splits the exact sequence {{EquationRef|1}}. A connection always exists.

Sometimes, this connection {{math|Γ}} is called the Ehresmann connection because it yields the horizontal distribution

: $\backslash mathrm\{H\}Y=\backslash Gamma\backslash left(Y\backslash times\_X\; \backslash mathrm\{T\}X\; \backslash right)\; \backslash subset\; \backslash mathrm\{T\}Y$

of {{math|TY}} and its horizontal decomposition {{math|TY {{=}} VY ⊕ HY}}.

At the same time, by an Ehresmann connection also is meant the following construction. Any connection {{math|Γ}} on a fibered manifold {{math|Y → X}} yields a horizontal lift {{math|Γ ∘ τ}} of a vector field {{mvar|τ}} on {{mvar|X}} onto {{mvar|Y}}, but need not defines the similar lift of a path in {{mvar|X}} into {{mvar|Y}}. Let

:

be two smooth paths in {{mvar|X}} and {{mvar|Y}}, respectively. Then {{math|t → y(t)}} is called the horizontal lift of {{math|x(t)}} if

:$\backslash pi(y(t))=\; x(t)\backslash ,,\; \backslash qquad\; \backslash dot\; y(t)\backslash in\; \backslash mathrm\{H\}Y\; \backslash ,,\; \backslash qquad\; t\backslash in\backslash mathbb\; R\backslash ,.$

A connection {{math|Γ}} is said to be the Ehresmann connection if, for each path {{math|x([0,1])}} in {{mvar|X}}, there exists its horizontal lift through any point {{math|y ∈ π⁻¹(x([0,1]))}}. A fibered manifold is a fiber bundle if and only if it admits such an Ehresmann connection.

Given a fibered manifold {{math|Y → X}}, let it be endowed with an atlas of fibered coordinates {{math|(x^μ, yⁱ)}}, and let {{math|Γ}} be a connection on {{math|Y → X}}. It yields uniquely the horizontal tangent-valued one-form

{{NumBlk|:|$\backslash Gamma\; =\; dx^\backslash mu\backslash otimes\; \backslash left(\backslash partial\_\backslash mu\; +\; \backslash Gamma\_\backslash mu^i\backslash left(x^\backslash nu,\; y^j\backslash right)\backslash partial\_i\backslash right)$|{{EquationRef|3}}}}

on {{mvar|Y}} which projects onto the canonical tangent-valued form (tautological one-form or solder form)

: $\backslash theta\_X=dx^\backslash mu\backslash otimes\backslash partial\_\backslash mu$

on {{mvar|X}}, and vice versa. With this form, the horizontal splitting {{EquationNote|2}} reads

: $\backslash Gamma:\backslash partial\_\backslash mu\backslash to\; \backslash partial\_\backslash mu\backslash rfloor\backslash Gamma=\backslash partial\_\backslash mu\; +\backslash Gamma^i\_\backslash mu\backslash partial\_i\backslash ,.$

In particular, the connection {{math|Γ}} in {{EquationNote|3}} yields the horizontal lift of any vector field {{math|τ {{=}} τ^μ ∂_μ}} on {{mvar|X}} to a projectable vector field

:$\backslash Gamma\; \backslash tau=\backslash tau\backslash rfloor\backslash Gamma=\backslash tau^\backslash mu\backslash left(\backslash partial\_\backslash mu\; +\backslash Gamma^i\_\backslash mu\backslash partial\_i\backslash right)\backslash subset\; \backslash mathrm\{H\}Y$

on {{mvar|Y}}.

The horizontal splitting {{EquationNote|2}} of the exact sequence {{EquationNote|1}} defines the corresponding splitting of the dual exact sequence

: $0\backslash to\; Y\backslash times\_X\; \backslash mathrm\{T\}^*X\; \backslash to\; \backslash mathrm\{T\}^*Y\backslash to\; \backslash mathrm\{V\}^*Y\backslash to\; 0\backslash ,,$

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

: $\backslash Gamma=\; \backslash left(dy^i\; -\backslash Gamma^i\_\backslash lambda\; dx^\backslash lambda\backslash right)\backslash otimes\backslash partial\_i\backslash ,,$

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 {{math|Y → X}}, let {{math|f : X′ → X}} be a morphism and {{math|f ∗ Y → X′}} the pullback bundle of {{mvar|Y}} by {{mvar|f}}. Then any connection {{math|Γ}} {{EquationNote|3}} on {{math|Y → X}} induces the pullback connection

: $f*\backslash Gamma=\backslash left(dy^i-\backslash left(\backslash Gamma\backslash circ\; \backslash tilde\; f\backslash right)^i\_\backslash lambda\backslash frac\{\backslash partial\; f^\backslash lambda\}\{\backslash partial\; x\text{'}^\backslash mu\}dx\text{'}^\backslash mu\backslash right)\backslash otimes\backslash partial\_i$

on {{math|f ∗ Y → X′}}.

Let {{math|J¹Y}} be the jet manifold of sections of a fibered manifold {{math|Y → X}}, with coordinates {{math|(x^μ, yⁱ, y{{su|p=i|b=μ}})}}. Due to the canonical imbedding

: $\backslash mathrm\{J\}^1Y\backslash to\_Y\; \backslash left(Y\backslash times\_X\; \backslash mathrm\{T\}^*X\; \backslash right)\backslash otimes\_Y\; \backslash mathrm\{T\}Y\backslash ,,\; \backslash qquad\; \backslash left(y^i\_\backslash mu\backslash right)\backslash to\; dx^\backslash mu\backslash otimes\; \backslash left(\backslash partial\_\backslash mu\; +\; y^i\_\backslash mu\backslash partial\_i\backslash right)\backslash ,,$

any connection {{math|Γ}} {{EquationNote|3}} on a fibered manifold {{math|Y → X}} is represented by a global section

: $\backslash Gamma\; :Y\backslash to\; \backslash mathrm\{J\}^1Y\backslash ,,\; \backslash qquad\; y\_\backslash lambda^i\backslash circ\backslash Gamma=\backslash Gamma\_\backslash lambda^i\backslash ,,$

of the jet bundle {{math|J¹Y → Y}}, and vice versa. It is an affine bundle modelled on a vector bundle

{{NumBlk|:|$\backslash left(Y\backslash times\_X\; T^*X\; \backslash right)\backslash otimes\_Y\; \backslash mathrm\{V\}Y\backslash to\; Y\backslash ,.$|{{EquationRef|4}}}}

There are the following corollaries of this fact.

{{ordered list|list_style_type=lower-roman

|Connections on a fibered manifold {{math|Y → X}} make up an affine space modelled on the vector space of soldering forms

{{NumBlk|:|$\backslash sigma=\backslash sigma^i\_\backslash mu\; dx^\backslash mu\backslash otimes\backslash partial\_i$|{{EquationRef|5}}}}

on {{math|Y → X}}, i.e., sections of the vector bundle {{EquationNote|4}}.

|Connection coefficients possess the coordinate transformation law

: $\{\backslash Gamma\text{'}\}^i\_\backslash lambda\; =\; \backslash frac\{\backslash partial\; x^\backslash mu\}\{\backslash partial\; \{x\text{'}\}^\backslash lambda\}\backslash left(\backslash partial\_\backslash mu\; \{y\text{'}\}^i+\backslash Gamma^j\_\backslash mu\backslash partial\_j\{y\text{'}\}^i\backslash right)\backslash ,.$

|Every connection {{math|Γ}} on a fibred manifold {{math|Y → X}} yields the first order differential operator

: $D\_\backslash Gamma:\backslash mathrm\{J\}^1Y\backslash to\_Y\; \backslash mathrm\{T\}^*X\backslash otimes\_Y\; \backslash mathrm\{V\}Y\backslash ,,\; \backslash qquad\; D\_\backslash Gamma\; =\; \backslash left(y^i\_\backslash lambda\; -\backslash Gamma^i\_\backslash lambda\backslash right)dx^\backslash lambda\backslash otimes\backslash partial\_i\backslash ,,$

on {{mvar|Y}} called the covariant differential relative to the connection {{math|Γ}}. If {{math|s : X → Y}} is a section, its covariant differential

: $\backslash nabla^\backslash Gamma\; s\; =\; \backslash left(\backslash partial\_\backslash lambda\; s^i\; -\; \backslash Gamma\_\backslash lambda^i\backslash circ\; s\backslash right)\; dx^\backslash lambda\backslash otimes\; \backslash partial\_i\backslash ,,$

and the covariant derivative

:$\backslash nabla\_\backslash tau^\backslash Gamma\; s=\backslash tau\backslash rfloor\backslash nabla^\backslash Gamma\; s$

along a vector field {{mvar|τ}} on {{mvar|X}} are defined.}}

Given the connection {{math|Γ}} {{EquationNote|3}} on a fibered manifold {{math|Y → X}}, its curvature is defined as the Nijenhuis differential

: $\backslash begin\{align\}$

R&=\tfrac12 d_\Gamma\Gamma\\&=\tfrac12 [\Gamma,\Gamma]_\mathrm{FN} \\&= \tfrac12 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\,.

\end{align}

This is a vertical-valued horizontal two-form on {{mvar|Y}}.

Given the connection {{math|Γ}} {{EquationNote|3}} and the soldering form {{mvar|σ}} {{EquationNote|5}}, a torsion of {{math|Γ}} with respect to {{mvar|σ}} is defined as

: $T\; =\; d\_\backslash Gamma\; \backslash sigma\; =\; \backslash left(\backslash partial\_\backslash lambda\backslash sigma\_\backslash mu^i\; +\; \backslash Gamma\_\backslash lambda^j\backslash partial\_j\backslash sigma\_\backslash mu^i\; -\backslash partial\_j\backslash Gamma\_\backslash lambda^i\backslash sigma\_\backslash mu^j\backslash right)\; \backslash ,\; dx^\backslash lambda\backslash wedge\; dx^\backslash mu\backslash otimes\; \backslash partial\_i\backslash ,.$

Let {{math|π : P → M}} be a principal bundle with a structure Lie group {{mvar|G}}. A principal connection on {{mvar|P}} usually is described by a Lie algebra-valued connection one-form on {{mvar|P}}. At the same time, a principal connection on {{mvar|P}} is a global section of the jet bundle {{math|J¹P → P}} which is equivariant with respect to the canonical right action of {{mvar|G}} in {{mvar|P}}. Therefore, it is represented by a global section of the quotient bundle {{math|C {{=}} J¹P/G → M}}, called the bundle of principal connections. It is an affine bundle modelled on the vector bundle {{math|VP/G → M}} whose typical fiber is the Lie algebra {{math|g}} of structure group {{mvar|G}}, and where {{mvar|G}} acts on by the adjoint representation. There is the canonical imbedding of {{mvar|C}} to the quotient bundle {{math|TP/G}} which also is called the bundle of principal connections.

Given a basis {{math|1={e_m}}} for a Lie algebra of {{mvar|G}}, the fiber bundle {{mvar|C}} is endowed with bundle coordinates {{math|(x^μ, a{{su|p=m|b=μ}})}}, and its sections are represented by vector-valued one-forms

: $A=dx^\backslash lambda\backslash otimes\; \backslash left(\backslash partial\_\backslash lambda\; +\; a^m\_\backslash lambda\; \{\backslash mathrm\; e\}\_m\backslash right)\backslash ,,$

where

: $a^m\_\backslash lambda\; \backslash ,\; dx^\backslash lambda\backslash otimes\; \{\backslash mathrm\; e\}\_m$

are the familiar local connection forms on {{mvar|M}}.

Let us note that the jet bundle {{math|J¹C}} of {{mvar|C}} is a configuration space of Yang–Mills gauge theory. It admits the canonical decomposition

:

&= \tfrac12\left(a_{\lambda\mu}^r + a_{\mu\lambda}^r - c_{pq}^r a_\lambda^p a_\mu^q\right) + \tfrac12\left(a_{\lambda\mu}^r - a_{\mu\lambda}^r + c_{pq}^r a_\lambda^p a_\mu^q\right)\,, \end{align}

where

: $F=\backslash tfrac\{1\}\{2\}\; F\_\{\backslash lambda\backslash mu\}^m\; \backslash ,\; dx^\backslash lambda\backslash wedge\; dx^\backslash mu\backslash otimes\; \{\backslash mathrm\; e\}\_m$

is called the strength form of a principal connection.

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

{{Reflist}}

• {{cite book|last1=Kolář|first1=Ivan|last2=Michor|first2=Peter|last3=Slovák|first3=Jan|url=http://www.emis.de/monographs/KSM/kmsbookh.pdf|title=Natural operators in differential geometry|year=1993|publisher=Springer-Verlag|access-date=2013-05-28|archive-url=https://web.archive.org/web/20170330154524/http://www.emis.de/monographs/KSM/kmsbookh.pdf|archive-date=2017-03-30|url-status=dead}}
• {{cite book|last1=Krupka|first1=Demeter|last2=Janyška|first2=Josef|title=Lectures on differential invariants|year=1990|publisher=Univerzita J. E. Purkyně v Brně|isbn=80-210-0165-8}}
• {{cite book|last1=Saunders|first1=D.J.|title=The geometry of jet bundles|url=https://archive.org/details/geometryofjetbun0000saun|url-access = registration|year=1989|publisher=Cambridge University Press|isbn=0-521-36948-7}}
• {{cite book|last1=Mangiarotti|first1=L.|author2-link=Gennadi Sardanashvily|last2=Sardanashvily|first2=G.|title=Connections in Classical and Quantum Field Theory|publisher=World Scientific|date=2000|isbn= 981-02-2013-8}}
• {{cite book|last=Sardanashvily|first=G.|author-link=Gennadi Sardanashvily|title=Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory|publisher=Lambert Academic Publishing|date=2013|isbn=978-3-659-37815-7|arxiv=0908.1886|bibcode=2009arXiv0908.1886S}}

{{Manifolds}}

Category:Connection (mathematics)

Category:Differential geometry

Category:Maps of manifolds

Category:Smooth functions