1. Formal definition
    Connection as a horizontal splitting  Connection as a tangent-valued form  Connection as a vertical-valued form  Connection as a jet bundle section 

2. Curvature and torsion

3. Bundle of principal connections

4. See also

5. Notes

6. References

In differential geometry, a fibered manifold is surjective submersion of smooth manifolds {{math|Y → 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 {{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}}.^[1]

Connection as a horizontal splitting

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

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

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

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

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.

Connection as a tangent-valued form

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

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

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

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

on {{mvar|Y}}.

Connection as a vertical-valued form

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

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

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

on {{math|f ∗ Y → X′}}.

Connection as a jet bundle section

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

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

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

There are the following corollaries of this fact.

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

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

and the covariant derivative

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

Curvature and torsion

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

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

Bundle of principal connections

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|{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

where

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

where

is called the strength form of a principal connection.

See also

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

Notes

References

• {{cite book|last1 = Kolář|first1=Ivan|last2=Michor|first2=Peter|last3=Slovák|first3=Jan|url=http://www.emis.de/monographs/KSM/kmsbookh.pdf|format=PDF|title=Natural operators in differential geometry|year = 1993|publisher = Springer-Verlag}}
• {{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|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|authorlink=Gennadi Sardanashvily|last=Sardanashvily |first=G. |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 }}

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