Frame bundle

<h2 id="definition-and-construction">Definition and construction</h2>
Let 
 
 
 
 E
 →
 X
 
 
 {\displaystyle E\to X}
 
 be a real <a href="/facts/Vector_bundle/cXACAa1I">vector bundle</a> of rank 
 
 
 
 k
 
 
 {\displaystyle k}
 
 over a <a href="/facts/Topological_space/v2ut7CbK">topological space</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
. A frame at a point 
 
 
 
 x
 ∈
 X
 
 
 {\displaystyle x\in X}
 
 is an <a href="/facts/Ordered_basis/89IPoN6c">ordered basis</a> for the vector space 
 
 
 
 
 E
 
 x
 
 
 
 
 {\displaystyle E_{x}}
 
. Equivalently, a frame can be viewed as a <a href="/facts/Linear_isomorphism/Q1PBKNVV">linear isomorphism</a>

p
        :
        
          
            R
          
          
            k
          
        
        →
        
          E
          
            x
          
        
        .
      
    
    {\displaystyle p:\mathbf {R} ^{k}\to E_{x}.}

The set of all frames at 
 
 
 
 x
 
 
 {\displaystyle x}
 
, denoted 
 
 
 
 
 F
 
 x
 
 
 
 
 {\displaystyle F_{x}}
 
, has a natural <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">right action</a> by the <a href="/facts/General_linear_group/B54gNILS">general linear group</a> 
 
 
 
 
 G
 L
 
 (
 k
 ,
 
 R
 
 )
 
 
 {\displaystyle \mathrm {GL} (k,\mathbb {R} )}
 
 of invertible 
 
 
 
 k
 ×
 k
 
 
 {\displaystyle k\times k}
 
 matrices: a group element 
 
 
 
 g
 ∈
 
 G
 L
 
 (
 k
 ,
 
 R
 
 )
 
 
 {\displaystyle g\in \mathrm {GL} (k,\mathbb {R} )}
 
 acts on the frame 
 
 
 
 p
 
 
 {\displaystyle p}
 
 via <a href="/facts/Function_composition/r5Pvnmth">composition</a> to give a new frame

p
        ∘
        g
        :
        
          
            R
          
          
            k
          
        
        →
        
          E
          
            x
          
        
        .
      
    
    {\displaystyle p\circ g:\mathbf {R} ^{k}\to E_{x}.}

This action of 
 
 
 
 
 G
 L
 
 (
 k
 ,
 
 R
 
 )
 
 
 {\displaystyle \mathrm {GL} (k,\mathbb {R} )}
 
 on 
 
 
 
 
 F
 
 x
 
 
 
 
 {\displaystyle F_{x}}
 
 is both <a href="/facts/Free_action/Vh9VtUUw">free</a> and <a href="/facts/Transitive_action/Vh9VtUUw">transitive</a> (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
 
 
 
 
 {\displaystyle F_{x}}
 
 is <a href="/facts/Homeomorphic/jMfWg3Mn">homeomorphic</a> to 
 
 
 
 
 G
 L
 
 (
 k
 ,
 
 R
 
 )
 
 
 {\displaystyle \mathrm {GL} (k,\mathbb {R} )}
 
 although it lacks a group structure, since there is no "preferred frame". The space 
 
 
 
 
 F
 
 x
 
 
 
 
 {\displaystyle F_{x}}
 
 is said to be a 
 
 
 
 
 G
 L
 
 (
 k
 ,
 
 R
 
 )
 
 
 {\displaystyle \mathrm {GL} (k,\mathbb {R} )}
 
-<a href="/facts/Torsor/N27maCdL">torsor</a>.
The frame bundle of 
 
 
 
 E
 
 
 {\displaystyle E}
 
, denoted by 
 
 
 
 F
 (
 E
 )
 
 
 {\displaystyle F(E)}
 
 or 
 
 
 
 
 F
 
 
 G
 L
 
 
 
 (
 E
 )
 
 
 {\displaystyle F_{\mathrm {GL} }(E)}
 
, is the <a href="/facts/Disjoint_union/x9QSfkr5">disjoint union</a> of all the 
 
 
 
 
 F
 
 x
 
 
 
 
 {\displaystyle F_{x}}
 
:

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

Each point in 
 
 
 
 F
 (
 E
 )
 
 
 {\displaystyle F(E)}
 
 is a pair (x, p) where 
 
 
 
 x
 
 
 {\displaystyle x}
 
 is a point in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 p
 
 
 {\displaystyle p}
 
 is a frame at 
 
 
 
 x
 
 
 {\displaystyle x}
 
. There is a natural projection 
 
 
 
 π
 :
 F
 (
 E
 )
 →
 X
 
 
 {\displaystyle \pi :F(E)\to X}
 
 which sends 
 
 
 
 (
 x
 ,
 p
 )
 
 
 {\displaystyle (x,p)}
 
 to 
 
 
 
 x
 
 
 {\displaystyle x}
 
. The group 
 
 
 
 
 G
 L
 
 (
 k
 ,
 
 R
 
 )
 
 
 {\displaystyle \mathrm {GL} (k,\mathbb {R} )}
 
 acts on 
 
 
 
 F
 (
 E
 )
 
 
 {\displaystyle F(E)}
 
 on the right as above. This action is clearly free and the <a href="/facts/Orbit_(group_theory)/Vh9VtUUw">orbits</a> are just the fibers of 
 
 
 
 π
 
 
 {\displaystyle \pi }
 
.

<h3>Principal bundle structure</h3>
The frame bundle 
 
 
 
 F
 (
 E
 )
 
 
 {\displaystyle F(E)}
 
 can be given a natural topology and bundle structure determined by that of 
 
 
 
 E
 
 
 {\displaystyle E}
 
. Let 
 
 
 
 (
 
 U
 
 i
 
 
 ,
 
 ϕ
 
 i
 
 
 )
 
 
 {\displaystyle (U_{i},\phi _{i})}
 
 be a <a href="/facts/Local_trivialization/SJ7Ek6cI">local trivialization</a> of 
 
 
 
 E
 
 
 {\displaystyle E}
 
. Then for each x ∈ Ui one has a linear isomorphism 
 
 
 
 
 ϕ
 
 i
 ,
 x
 
 
 :
 
 E
 
 x
 
 
 →
 
 
 R
 
 
 k
 
 
 
 
 {\displaystyle \phi _{i,x}:E_{x}\to \mathbb {R} ^{k}}
 
. This data determines a bijection

ψ
          
            i
          
        
        :
        
          π
          
            −
            1
          
        
        (
        
          U
          
            i
          
        
        )
        →
        
          U
          
            i
          
        
        ×
        
          G
          L
        
        (
        k
        ,
        
          R
        
        )
      
    
    {\displaystyle \psi _{i}:\pi ^{-1}(U_{i})\to U_{i}\times \mathrm {GL} (k,\mathbb {R} )}

given by

ψ
          
            i
          
        
        (
        x
        ,
        p
        )
        =
        (
        x
        ,
        
          ϕ
          
            i
            ,
            x
          
        
        ∘
        p
        )
        .
      
    
    {\displaystyle \psi _{i}(x,p)=(x,\phi _{i,x}\circ p).}

With these bijections, each 
 
 
 
 
 π
 
 −
 1
 
 
 (
 
 U
 
 i
 
 
 )
 
 
 {\displaystyle \pi ^{-1}(U_{i})}
 
 can be given the topology of 
 
 
 
 
 U
 
 i
 
 
 ×
 
 G
 L
 
 (
 k
 ,
 
 R
 
 )
 
 
 {\displaystyle U_{i}\times \mathrm {GL} (k,\mathbb {R} )}
 
. The topology on 
 
 
 
 F
 (
 E
 )
 
 
 {\displaystyle F(E)}
 
 is the <a href="/facts/Final_topology/gk2UV5va">final topology</a> coinduced by the inclusion maps 
 
 
 
 
 π
 
 −
 1
 
 
 (
 
 U
 
 i
 
 
 )
 →
 F
 (
 E
 )
 
 
 {\displaystyle \pi ^{-1}(U_{i})\to F(E)}
 
.
With all of the above data the frame bundle 
 
 
 
 F
 (
 E
 )
 
 
 {\displaystyle F(E)}
 
 becomes a <a href="/facts/Principal_fiber_bundle/bbB2D94B">principal fiber bundle</a> over 
 
 
 
 X
 
 
 {\displaystyle X}
 
 with <a href="/facts/Structure_group/SJ7Ek6cI">structure group</a> 
 
 
 
 
 G
 L
 
 (
 k
 ,
 
 R
 
 )
 
 
 {\displaystyle \mathrm {GL} (k,\mathbb {R} )}
 
 and local trivializations 
 
 
 
 (
 {
 
 U
 
 i
 
 
 }
 ,
 {
 
 ψ
 
 i
 
 
 }
 )
 
 
 {\displaystyle (\{U_{i}\},\{\psi _{i}\})}
 
. One can check that the <a href="/facts/Transition_map/pjoya29t">transition functions</a> of 
 
 
 
 F
 (
 E
 )
 
 
 {\displaystyle F(E)}
 
 are the same as those of 
 
 
 
 E
 
 
 {\displaystyle E}
 
.
The above all works in the smooth category as well: if 
 
 
 
 E
 
 
 {\displaystyle E}
 
 is a smooth vector bundle over a <a href="/facts/Smooth_manifold/aSnbXKcV">smooth manifold</a> 
 
 
 
 M
 
 
 {\displaystyle M}
 
 then the frame bundle of 
 
 
 
 E
 
 
 {\displaystyle E}
 
 can be given the structure of a smooth principal bundle over 
 
 
 
 M
 
 
 {\displaystyle M}
 
.

<h2 id="associated-vector-bundles">Associated vector bundles</h2>
A vector bundle 
 
 
 
 E
 
 
 {\displaystyle E}
 
 and its frame bundle 
 
 
 
 F
 (
 E
 )
 
 
 {\displaystyle F(E)}
 
 are <a href="/facts/Associated_bundle/7JfKDcBJ">associated bundles</a>. Each one determines the other. The frame bundle 
 
 
 
 F
 (
 E
 )
 
 
 {\displaystyle F(E)}
 
 can be constructed from 
 
 
 
 E
 
 
 {\displaystyle E}
 
 as above, or more abstractly using the <a href="/facts/Fiber_bundle_construction_theorem/szr3qWYC">fiber bundle construction theorem</a>. With the latter method, 
 
 
 
 F
 (
 E
 )
 
 
 {\displaystyle F(E)}
 
 is the fiber bundle with same base, structure group, trivializing neighborhoods, and transition functions as 
 
 
 
 E
 
 
 {\displaystyle E}
 
 but with abstract fiber 
 
 
 
 
 G
 L
 
 (
 k
 ,
 
 R
 
 )
 
 
 {\displaystyle \mathrm {GL} (k,\mathbb {R} )}
 
, where the action of structure group 
 
 
 
 
 G
 L
 
 (
 k
 ,
 
 R
 
 )
 
 
 {\displaystyle \mathrm {GL} (k,\mathbb {R} )}
 
 on the fiber 
 
 
 
 
 G
 L
 
 (
 k
 ,
 
 R
 
 )
 
 
 {\displaystyle \mathrm {GL} (k,\mathbb {R} )}
 
 is that of left multiplication.
Given any <a href="/facts/Linear_representation/3ydLtaGH">linear representation</a> 
 
 
 
 ρ
 :
 
 G
 L
 
 (
 k
 ,
 
 R
 
 )
 →
 
 G
 L
 
 (
 V
 ,
 
 F
 
 )
 
 
 {\displaystyle \rho :\mathrm {GL} (k,\mathbb {R} )\to \mathrm {GL} (V,\mathbb {F} )}
 
 there is a vector bundle

F
        
        (
        E
        )
        
          ×
          
            ρ
          
        
        V
      
    
    {\displaystyle \mathrm {F} (E)\times _{\rho }V}

associated with 
 
 
 
 F
 (
 E
 )
 
 
 {\displaystyle F(E)}
 
 which is given by product 
 
 
 
 F
 (
 E
 )
 ×
 V
 
 
 {\displaystyle F(E)\times V}
 
 modulo the <a href="/facts/Equivalence_relation/1MQ5rtkW">equivalence relation</a> 
 
 
 
 (
 p
 g
 ,
 v
 )
 ∼
 (
 p
 ,
 ρ
 (
 g
 )
 v
 )
 
 
 {\displaystyle (pg,v)\sim (p,\rho (g)v)}
 
 for all 
 
 
 
 g
 
 
 {\displaystyle g}
 
 in 
 
 
 
 
 G
 L
 
 (
 k
 ,
 
 R
 
 )
 
 
 {\displaystyle \mathrm {GL} (k,\mathbb {R} )}
 
. Denote the equivalence classes by 
 
 
 
 [
 p
 ,
 v
 ]
 
 
 {\displaystyle [p,v]}
 
.
The vector bundle 
 
 
 
 E
 
 
 {\displaystyle E}
 
 is <a href="/facts/Naturally_isomorphic/xwPumTHt">naturally isomorphic</a> to the bundle 
 
 
 
 F
 (
 E
 )
 
 ×
 
 ρ
 
 
 
 
 R
 
 
 k
 
 
 
 
 {\displaystyle F(E)\times _{\rho }\mathbb {R} ^{k}}
 
 where 
 
 
 
 ρ
 
 
 {\displaystyle \rho }
 
 is the fundamental representation of 
 
 
 
 
 G
 L
 
 (
 k
 ,
 
 R
 
 )
 
 
 {\displaystyle \mathrm {GL} (k,\mathbb {R} )}
 
 on 
 
 
 
 
 
 R
 
 
 k
 
 
 
 
 {\displaystyle \mathbb {R} ^{k}}
 
. The isomorphism is given by

[
        p
        ,
        v
        ]
        ↦
        p
        (
        v
        )
      
    
    {\displaystyle [p,v]\mapsto p(v)}

where 
 
 
 
 v
 
 
 {\displaystyle v}
 
 is a vector in 
 
 
 
 
 
 R
 
 
 k
 
 
 
 
 {\displaystyle \mathbb {R} ^{k}}
 
 and 
 
 
 
 p
 :
 
 
 R
 
 
 k
 
 
 →
 
 E
 
 x
 
 
 
 
 {\displaystyle p:\mathbb {R} ^{k}\to E_{x}}
 
 is a frame at 
 
 
 
 x
 
 
 {\displaystyle x}
 
. One can easily check that this map is <a href="/facts/Well-defined/GAc0XbWz">well-defined</a>.
Any vector bundle associated with 
 
 
 
 E
 
 
 {\displaystyle E}
 
 can be given by the above construction. For example, the <a href="/facts/Dual_bundle/cf9ATzjb">dual bundle</a> of 
 
 
 
 E
 
 
 {\displaystyle E}
 
 is given by 
 
 
 
 F
 (
 E
 )
 
 ×
 
 
 ρ
 
 ∗
 
 
 
 
 (
 
 
 R
 
 
 k
 
 
 
 )
 
 ∗
 
 
 
 
 {\displaystyle F(E)\times _{\rho ^{*}}(\mathbb {R} ^{k})^{*}}
 
 where 
 
 
 
 
 ρ
 
 ∗
 
 
 
 
 {\displaystyle \rho ^{*}}
 
 is the <a href="/facts/Dual_representation/HJWTCk5d">dual</a> of the fundamental representation. <a href="/facts/Tensor_bundle/cAC6F6rF">Tensor bundles</a> of 
 
 
 
 E
 
 
 {\displaystyle E}
 
 can be constructed in a similar manner.

<h2 id="tangent-frame-bundle">Tangent frame bundle</h2>
The tangent frame bundle (or simply the frame bundle) of a <a href="/facts/Smooth_manifold/aSnbXKcV">smooth manifold</a> 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is the frame bundle associated with the <a href="/facts/Tangent_bundle/dq8zW9Kb">tangent bundle</a> of 
 
 
 
 M
 
 
 {\displaystyle M}
 
. The frame bundle of 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is often denoted 
 
 
 
 F
 M
 
 
 {\displaystyle FM}
 
 or 
 
 
 
 
 G
 L
 
 (
 M
 )
 
 
 {\displaystyle \mathrm {GL} (M)}
 
 rather than 
 
 
 
 F
 (
 T
 M
 )
 
 
 {\displaystyle F(TM)}
 
. In physics, it is sometimes denoted 
 
 
 
 L
 M
 
 
 {\displaystyle LM}
 
. If 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is 
 
 
 
 n
 
 
 {\displaystyle n}
 
-dimensional then the tangent bundle has rank 
 
 
 
 n
 
 
 {\displaystyle n}
 
, so the frame bundle of 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is a principal 
 
 
 
 
 G
 L
 
 (
 n
 ,
 
 R
 
 )
 
 
 {\displaystyle \mathrm {GL} (n,\mathbb {R} )}
 
 bundle over 
 
 
 
 M
 
 
 {\displaystyle M}
 
.

<h3>Smooth frames</h3>
<a href="/facts/Section_(fiber_bundle)/DLnpXS5a">Local sections</a> of the frame bundle of 
 
 
 
 M
 
 
 {\displaystyle M}
 
 are called <a href="/facts/Smooth_frame/3jNPLngL">smooth frames</a> on 
 
 
 
 M
 
 
 {\displaystyle M}
 
. The cross-section theorem for principal bundles states that the frame bundle is trivial over any open set in 
 
 
 
 U
 
 
 {\displaystyle U}
 
 in 
 
 
 
 M
 
 
 {\displaystyle M}
 
 which admits a smooth frame. Given a smooth frame 
 
 
 
 s
 :
 U
 →
 F
 U
 
 
 {\displaystyle s:U\to FU}
 
, the trivialization 
 
 
 
 ψ
 :
 F
 U
 →
 U
 ×
 
 G
 L
 
 (
 n
 ,
 
 R
 
 )
 
 
 {\displaystyle \psi :FU\to U\times \mathrm {GL} (n,\mathbb {R} )}
 
 is given by

ψ
        (
        p
        )
        =
        (
        x
        ,
        s
        (
        x
        
          )
          
            −
            1
          
        
        ∘
        p
        )
      
    
    {\displaystyle \psi (p)=(x,s(x)^{-1}\circ p)}

where 
 
 
 
 p
 
 
 {\displaystyle p}
 
 is a frame at 
 
 
 
 x
 
 
 {\displaystyle x}
 
. It follows that a manifold is <a href="/facts/Parallelizable_manifold/CQq7nQsJ">parallelizable</a> if and only if the frame bundle of 
 
 
 
 M
 
 
 {\displaystyle M}
 
 admits a global section.
Since the tangent bundle of 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is trivializable over coordinate neighborhoods of 
 
 
 
 M
 
 
 {\displaystyle M}
 
 so is the frame bundle. In fact, given any coordinate neighborhood 
 
 
 
 U
 
 
 {\displaystyle U}
 
 with coordinates 
 
 
 
 (
 
 x
 
 1
 
 
 ,
 …
 ,
 
 x
 
 n
 
 
 )
 
 
 {\displaystyle (x^{1},\ldots ,x^{n})}
 
 the coordinate vector fields

(
          
            
              
                ∂
                
                  ∂
                  
                    x
                    
                      1
                    
                  
                
              
            
            ,
            …
            ,
            
              
                ∂
                
                  ∂
                  
                    x
                    
                      n
                    
                  
                
              
            
          
          )
        
      
    
    {\displaystyle \left({\frac {\partial }{\partial x^{1}}},\ldots ,{\frac {\partial }{\partial x^{n}}}\right)}

define a smooth frame on 
 
 
 
 U
 
 
 {\displaystyle 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 <a href="/facts/Method_of_moving_frames/3jNPLngL">method of moving frames</a>.

<h3>Solder form</h3>
The frame bundle of a manifold 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is a special type of principal bundle in the sense that its geometry is fundamentally tied to the geometry of 
 
 
 
 M
 
 
 {\displaystyle M}
 
. This relationship can be expressed by means of a <a href="/facts/Vector-valued_differential_form/GohcjkUh">vector-valued 1-form</a> on 
 
 
 
 F
 M
 
 
 {\displaystyle FM}
 
 called the <a href="/facts/Solder_form/1zNxoKvT">solder form</a> (also known as the fundamental or <a href="/facts/Tautological_one-form/FH8WKyxV">tautological 1-form</a>). Let 
 
 
 
 x
 
 
 {\displaystyle x}
 
 be a point of the manifold 
 
 
 
 M
 
 
 {\displaystyle M}
 
 and 
 
 
 
 p
 
 
 {\displaystyle p}
 
 a frame at 
 
 
 
 x
 
 
 {\displaystyle x}
 
, so that

p
        :
        
          
            R
          
          
            n
          
        
        →
        
          T
          
            x
          
        
        M
      
    
    {\displaystyle p:\mathbf {R} ^{n}\to T_{x}M}

is a linear isomorphism of 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
 with the tangent space of 
 
 
 
 M
 
 
 {\displaystyle M}
 
 at 
 
 
 
 x
 
 
 {\displaystyle x}
 
. The solder form of 
 
 
 
 F
 M
 
 
 {\displaystyle FM}
 
 is the 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
-valued 1-form 
 
 
 
 θ
 
 
 {\displaystyle \theta }
 
 defined by

θ
          
            p
          
        
        (
        ξ
        )
        =
        
          p
          
            −
            1
          
        
        
          d
        
        π
        (
        ξ
        )
      
    
    {\displaystyle \theta _{p}(\xi )=p^{-1}\mathrm {d} \pi (\xi )}

where ξ is a tangent vector to 
 
 
 
 F
 M
 
 
 {\displaystyle FM}
 
 at the point 
 
 
 
 (
 x
 ,
 p
 )
 
 
 {\displaystyle (x,p)}
 
, and 
 
 
 
 
 p
 
 −
 1
 
 
 :
 
 T
 
 x
 
 
 M
 →
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle p^{-1}:T_{x}M\to \mathbb {R} ^{n}}
 
 is the inverse of the frame map, and 
 
 
 
 d
 π
 
 
 {\displaystyle d\pi }
 
 is the <a href="/facts/Pushforward_(differential)/xLsoLwVH">differential</a> of the projection map 
 
 
 
 π
 :
 F
 M
 →
 M
 
 
 {\displaystyle \pi :FM\to M}
 
. The solder form is horizontal in the sense that it vanishes on vectors tangent to the fibers of 
 
 
 
 π
 
 
 {\displaystyle \pi }
 
 and <a href="/facts/Equivariant/KkIkDwIH">right equivariant</a> in the sense that

R
          
            g
          
          
            ∗
          
        
        θ
        =
        
          g
          
            −
            1
          
        
        θ
      
    
    {\displaystyle R_{g}^{*}\theta =g^{-1}\theta }

where 
 
 
 
 
 R
 
 g
 
 
 
 
 {\displaystyle R_{g}}
 
 is right translation by 
 
 
 
 g
 ∈
 
 G
 L
 
 (
 n
 ,
 
 R
 
 )
 
 
 {\displaystyle g\in \mathrm {GL} (n,\mathbb {R} )}
 
. A form with these properties is called a basic or <a href="/facts/Tensorial_form/GohcjkUh">tensorial form</a> on 
 
 
 
 F
 M
 
 
 {\displaystyle FM}
 
. Such forms are in 1-1 correspondence with 
 
 
 
 T
 M
 
 
 {\displaystyle TM}
 
-valued 1-forms on 
 
 
 
 M
 
 
 {\displaystyle M}
 
 which are, in turn, in 1-1 correspondence with smooth <a href="/facts/Bundle_map/pfp2zwxo">bundle maps</a> 
 
 
 
 T
 M
 →
 T
 M
 
 
 {\displaystyle TM\to TM}
 
 over 
 
 
 
 M
 
 
 {\displaystyle M}
 
. Viewed in this light 
 
 
 
 θ
 
 
 {\displaystyle \theta }
 
 is just the <a href="/facts/Identity_function/9AtsP0LC">identity map</a> on 
 
 
 
 T
 M
 
 
 {\displaystyle TM}
 
.
As a naming convention, the term "tautological one-form" is usually reserved for the case where the form has a canonical definition, as it does here, while "solder form" is more appropriate for those cases where the form is not canonically defined. This convention is not being observed here.

<h2 id="orthonormal-frame-bundle">Orthonormal frame bundle</h2>
If a vector bundle 
 
 
 
 E
 
 
 {\displaystyle E}
 
 is equipped with a <a href="/facts/Riemannian_bundle_metric/PLoXIrWV">Riemannian bundle metric</a> then each fiber 
 
 
 
 
 E
 
 x
 
 
 
 
 {\displaystyle E_{x}}
 
 is not only a vector space but an <a href="/facts/Inner_product_space/HoyjElEy">inner product space</a>. It is then possible to talk about the set of all <a href="/facts/Orthonormal_frame/lQFvBnQq">orthonormal frames</a> for 
 
 
 
 
 E
 
 x
 
 
 
 
 {\displaystyle E_{x}}
 
. An orthonormal frame for 
 
 
 
 
 E
 
 x
 
 
 
 
 {\displaystyle E_{x}}
 
 is an ordered <a href="/facts/Orthonormal_basis/pzfKHlJG">orthonormal basis</a> for 
 
 
 
 
 E
 
 x
 
 
 
 
 {\displaystyle E_{x}}
 
, or, equivalently, a <a href="/facts/Linear_isometry/K5wX8ah6">linear isometry</a>

p
        :
        
          
            R
          
          
            k
          
        
        →
        
          E
          
            x
          
        
      
    
    {\displaystyle p:\mathbb {R} ^{k}\to E_{x}}

where 
 
 
 
 
 
 R
 
 
 k
 
 
 
 
 {\displaystyle \mathbb {R} ^{k}}
 
 is equipped with the standard <a href="/facts/Euclidean_metric/9qDoQKQe">Euclidean metric</a>. The <a href="/facts/Orthogonal_group/jKWM1KDM">orthogonal group</a> 
 
 
 
 
 O
 
 (
 k
 )
 
 
 {\displaystyle \mathrm {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
 )
 
 
 {\displaystyle \mathrm {O} (k)}
 
-<a href="/facts/Torsor/N27maCdL">torsor</a>.
The orthonormal frame bundle of 
 
 
 
 E
 
 
 {\displaystyle E}
 
, denoted 
 
 
 
 
 F
 
 
 O
 
 
 
 (
 E
 )
 
 
 {\displaystyle F_{\mathrm {O} }(E)}
 
, is the set of all orthonormal frames at each point 
 
 
 
 x
 
 
 {\displaystyle x}
 
 in the base space 
 
 
 
 X
 
 
 {\displaystyle 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
 
 
 {\displaystyle k}
 
 Riemannian vector bundle 
 
 
 
 E
 →
 X
 
 
 {\displaystyle E\to X}
 
 is a principal 
 
 
 
 
 O
 
 (
 k
 )
 
 
 {\displaystyle \mathrm {O} (k)}
 
-bundle over 
 
 
 
 X
 
 
 {\displaystyle X}
 
. Again, the construction works just as well in the smooth category.
If the vector bundle 
 
 
 
 E
 
 
 {\displaystyle E}
 
 is <a href="/facts/Orientability/1SEQM2Pt">orientable</a> then one can define the oriented orthonormal frame bundle of 
 
 
 
 E
 
 
 {\displaystyle E}
 
, denoted 
 
 
 
 
 F
 
 
 S
 O
 
 
 
 (
 E
 )
 
 
 {\displaystyle F_{\mathrm {SO} }(E)}
 
, as the principal 
 
 
 
 
 S
 O
 
 (
 k
 )
 
 
 {\displaystyle \mathrm {SO} (k)}
 
-bundle of all positively oriented orthonormal frames.
If 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is an 
 
 
 
 n
 
 
 {\displaystyle n}
 
-dimensional <a href="/facts/Riemannian_manifold/5KYnmEgF">Riemannian manifold</a>, then the orthonormal frame bundle of 
 
 
 
 M
 
 
 {\displaystyle M}
 
, denoted 
 
 
 
 
 F
 
 
 O
 
 
 
 (
 M
 )
 
 
 {\displaystyle F_{\mathrm {O} }(M)}
 
 or 
 
 
 
 
 O
 
 (
 M
 )
 
 
 {\displaystyle \mathrm {O} (M)}
 
, is the orthonormal frame bundle associated with the tangent bundle of 
 
 
 
 M
 
 
 {\displaystyle M}
 
 (which is equipped with a Riemannian metric by definition). If 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is orientable, then one also has the oriented orthonormal frame bundle 
 
 
 
 
 F
 
 
 S
 O
 
 
 
 M
 
 
 {\displaystyle F_{\mathrm {SO} }M}
 
.
Given a Riemannian vector bundle 
 
 
 
 E
 
 
 {\displaystyle E}
 
, the orthonormal frame bundle is a principal 
 
 
 
 
 O
 
 (
 k
 )
 
 
 {\displaystyle \mathrm {O} (k)}
 
-<a href="/facts/Subbundle/KFWrEleF">subbundle</a> of the general linear frame bundle. In other words, the inclusion map

i
        :
        
          
            
              F
            
          
          
            
              O
            
          
        
        (
        E
        )
        →
        
          
            
              F
            
          
          
            
              G
              L
            
          
        
        (
        E
        )
      
    
    {\displaystyle i:{\mathrm {F} }_{\mathrm {O} }(E)\to {\mathrm {F} }_{\mathrm {GL} }(E)}

is principal <a href="/facts/Bundle_map/pfp2zwxo">bundle map</a>. One says that 
 
 
 
 
 F
 
 
 O
 
 
 
 (
 E
 )
 
 
 {\displaystyle F_{\mathrm {O} }(E)}
 
 is a <a href="/facts/Reduction_of_the_structure_group/NmGVT6v0">reduction of the structure group</a> of 
 
 
 
 
 F
 
 
 G
 L
 
 
 
 (
 E
 )
 
 
 {\displaystyle F_{\mathrm {GL} }(E)}
 
 from 
 
 
 
 
 G
 L
 
 (
 n
 ,
 
 R
 
 )
 
 
 {\displaystyle \mathrm {GL} (n,\mathbb {R} )}
 
 to 
 
 
 
 
 O
 
 (
 k
 )
 
 
 {\displaystyle \mathrm {O} (k)}
 
.

<h2 id="g-structures">G-structures</h2>
See also: <a href="/facts/G-structure/NmGVT6v0">G-structure</a>
If a smooth manifold 
 
 
 
 M
 
 
 {\displaystyle M}
 
 comes with additional structure it is often natural to consider a subbundle of the full frame bundle of 
 
 
 
 M
 
 
 {\displaystyle M}
 
 which is adapted to the given structure. For example, if 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is a Riemannian manifold we saw above that it is natural to consider the orthonormal frame bundle of 
 
 
 
 M
 
 
 {\displaystyle M}
 
. The orthonormal frame bundle is just a reduction of the structure group of 
 
 
 
 
 F
 
 
 G
 L
 
 
 
 (
 M
 )
 
 
 {\displaystyle F_{\mathrm {GL} }(M)}
 
 to the orthogonal group 
 
 
 
 
 O
 
 (
 n
 )
 
 
 {\displaystyle \mathrm {O} (n)}
 
.
In general, if 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is a smooth 
 
 
 
 n
 
 
 {\displaystyle n}
 
-manifold and 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is a <a href="/facts/Lie_subgroup/a9jCQyjF">Lie subgroup</a> of 
 
 
 
 
 G
 L
 
 (
 n
 ,
 
 R
 
 )
 
 
 {\displaystyle \mathrm {GL} (n,\mathbb {R} )}
 
 we define a <a href="/facts/G-structure/NmGVT6v0">G-structure</a> on 
 
 
 
 M
 
 
 {\displaystyle M}
 
 to be a <a href="/facts/Reduction_of_the_structure_group/NmGVT6v0">reduction of the structure group</a> of 
 
 
 
 
 F
 
 
 G
 L
 
 
 
 (
 M
 )
 
 
 {\displaystyle F_{\mathrm {GL} }(M)}
 
 to 
 
 
 
 G
 
 
 {\displaystyle G}
 
. Explicitly, this is a principal 
 
 
 
 G
 
 
 {\displaystyle G}
 
-bundle 
 
 
 
 
 F
 
 G
 
 
 (
 M
 )
 
 
 {\displaystyle F_{G}(M)}
 
 over 
 
 
 
 M
 
 
 {\displaystyle M}
 
 together with a 
 
 
 
 G
 
 
 {\displaystyle G}
 
-equivariant <a href="/facts/Bundle_map/pfp2zwxo">bundle map</a>

F
            
          
          
            G
          
        
        (
        M
        )
        →
        
          
            
              F
            
          
          
            
              G
              L
            
          
        
        (
        M
        )
      
    
    {\displaystyle {\mathrm {F} }_{G}(M)\to {\mathrm {F} }_{\mathrm {GL} }(M)}

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

<ul><li>Every <a href="/facts/Orientability/1SEQM2Pt">oriented manifold</a> has an oriented frame bundle which is just a 
 
 
 
 
 
 G
 L
 
 
 +
 
 
 (
 n
 ,
 
 R
 
 )
 
 
 {\displaystyle \mathrm {GL} ^{+}(n,\mathbb {R} )}
 
-structure on 
 
 
 
 M
 
 
 {\displaystyle M}
 
.</li>
<li>A <a href="/facts/Volume_form/2x37pDLg">volume form</a> on 
 
 
 
 M
 
 
 {\displaystyle M}
 
 determines a 
 
 
 
 
 S
 L
 
 (
 n
 ,
 
 R
 
 )
 
 
 {\displaystyle \mathrm {SL} (n,\mathbb {R} )}
 
-structure on 
 
 
 
 M
 
 
 {\displaystyle M}
 
.</li>
<li>A 
 
 
 
 2
 n
 
 
 {\displaystyle 2n}
 
-dimensional <a href="/facts/Symplectic_manifold/JvNrJkC7">symplectic manifold</a> has a natural 
 
 
 
 
 S
 p
 
 (
 2
 n
 ,
 
 R
 
 )
 
 
 {\displaystyle \mathrm {Sp} (2n,\mathbb {R} )}
 
-structure.</li>
<li>A 
 
 
 
 2
 n
 
 
 {\displaystyle 2n}
 
-dimensional <a href="/facts/Complex_manifold/DlvPnpzm">complex</a> or <a href="/facts/Almost_complex_manifold/wsXTsuC6">almost complex manifold</a> has a natural 
 
 
 
 
 G
 L
 
 (
 n
 ,
 
 C
 
 )
 
 
 {\displaystyle \mathrm {GL} (n,\mathbb {C} )}
 
-structure.</li></ul>
In many of these instances, a 
 
 
 
 G
 
 
 {\displaystyle G}
 
-structure on 
 
 
 
 M
 
 
 {\displaystyle M}
 
 uniquely determines the corresponding structure on 
 
 
 
 M
 
 
 {\displaystyle M}
 
. For example, a 
 
 
 
 
 S
 L
 
 (
 n
 ,
 
 R
 
 )
 
 
 {\displaystyle \mathrm {SL} (n,\mathbb {R} )}
 
-structure on 
 
 
 
 M
 
 
 {\displaystyle M}
 
 determines a volume form on 
 
 
 
 M
 
 
 {\displaystyle M}
 
. However, in some cases, such as for symplectic and complex manifolds, an added <a href="/facts/Integrability_condition/PKp57zwz">integrability condition</a> is needed. A 
 
 
 
 
 S
 p
 
 (
 2
 n
 ,
 
 R
 
 )
 
 
 {\displaystyle \mathrm {Sp} (2n,\mathbb {R} )}
 
-structure on 
 
 
 
 M
 
 
 {\displaystyle M}
 
 uniquely determines a <a href="/facts/Nondegenerate_form/9TLGBvSH">nondegenerate</a> <a href="/facts/2-form/T9gyHtMv">2-form</a> on 
 
 
 
 M
 
 
 {\displaystyle M}
 
, but for 
 
 
 
 M
 
 
 {\displaystyle M}
 
 to be symplectic, this 2-form must also be <a href="/facts/Closed_differential_form/uvEUbzUK">closed</a>.

<ul><li>Kobayashi, Shoshichi; Nomizu, Katsumi (1996), <a href="/facts/Foundations_of_Differential_Geometry/lgGBjhhd">Foundations of Differential Geometry</a>, vol. 1 (New ed.), <a href="/facts/Wiley_Interscience/53g3LqJc">Wiley Interscience</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-15733-3</li>
<li>Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), <a href="https://web.archive.org/web/20170330154524/http://www.emis.de/monographs/KSM/kmsbookh.pdf">Natural operators in differential geometry</a> (PDF), Springer-Verlag, archived from <a href="http://www.emis.de/monographs/KSM/kmsbookh.pdf">the original</a> (PDF) on 2017-03-30, retrieved 2008-08-02</li>
<li><a href="/facts/Shlomo_Sternberg/IsGM8VzY">Sternberg, S.</a> (1983), Lectures on Differential Geometry ((2nd ed.) ed.), New York: Chelsea Publishing Co., <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-1385-4</li></ul>

Frame bundle open-in-new

Frame bundle