General covariant transformations

<h2 id="mathematical-definition">Mathematical definition</h2>
Let 
 
 
 
 π
 :
 Y
 →
 X
 
 
 {\displaystyle \pi :Y\to X}
 
 be a <a href="/facts/Fibered_manifold/4g6exYHA">fibered manifold</a> with local fibered coordinates 
 
 
 
 (
 
 x
 
 λ
 
 
 ,
 
 y
 
 i
 
 
 )
 
 
 
 {\displaystyle (x^{\lambda },y^{i})\,}
 
. Every automorphism of 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 is projected onto a <a href="/facts/Diffeomorphism/855N5YYj">diffeomorphism</a> of its base 
 
 
 
 X
 
 
 {\displaystyle X}
 
. However, the converse is not true. A diffeomorphism of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 need not give rise to an automorphism of 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
.
In particular, an <a href="/facts/Lie_group/a9jCQyjF">infinitesimal generator</a> of a one-parameter <a href="/facts/Lie_group/a9jCQyjF">Lie group</a> of automorphisms of 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 is a projectable <a href="/facts/Vector_field/ccFNuPPp">vector field</a>

u
        =
        
          u
          
            λ
          
        
        (
        
          x
          
            μ
          
        
        )
        
          ∂
          
            λ
          
        
        +
        
          u
          
            i
          
        
        (
        
          x
          
            μ
          
        
        ,
        
          y
          
            j
          
        
        )
        
          ∂
          
            i
          
        
      
    
    {\displaystyle u=u^{\lambda }(x^{\mu })\partial _{\lambda }+u^{i}(x^{\mu },y^{j})\partial _{i}}

on 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
. This vector field is projected onto a vector field 
 
 
 
 τ
 =
 
 u
 
 λ
 
 
 
 ∂
 
 λ
 
 
 
 
 {\displaystyle \tau =u^{\lambda }\partial _{\lambda }}
 
 on 
 
 
 
 X
 
 
 {\displaystyle X}
 
, whose flow is a one-parameter group of diffeomorphisms of 
 
 
 
 X
 
 
 {\displaystyle X}
 
. Conversely, let 
 
 
 
 τ
 =
 
 τ
 
 λ
 
 
 
 ∂
 
 λ
 
 
 
 
 {\displaystyle \tau =\tau ^{\lambda }\partial _{\lambda }}
 
 be a vector field on 
 
 
 
 X
 
 
 {\displaystyle X}
 
. There is a problem of constructing its lift to a projectable vector field on 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 projected onto 
 
 
 
 τ
 
 
 {\displaystyle \tau }
 
. Such a lift always exists, but it need not be canonical. Given a <a href="/facts/Connection_(fibred_manifold)/QCKqyt2Y">connection</a> 
 
 
 
 Γ
 
 
 {\displaystyle \Gamma }
 
 on 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
, every vector field 
 
 
 
 τ
 
 
 {\displaystyle \tau }
 
 on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 gives rise to the horizontal vector field

Γ
        τ
        =
        
          τ
          
            λ
          
        
        (
        
          ∂
          
            λ
          
        
        +
        
          Γ
          
            λ
          
          
            i
          
        
        
          ∂
          
            i
          
        
        )
      
    
    {\displaystyle \Gamma \tau =\tau ^{\lambda }(\partial _{\lambda }+\Gamma _{\lambda }^{i}\partial _{i})}

on 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
. This horizontal lift 
 
 
 
 τ
 →
 Γ
 τ
 
 
 {\displaystyle \tau \to \Gamma \tau }
 
 yields a <a href="/facts/Monomorphism/Q9BrtDTs">monomorphism</a> of the 
 
 
 
 
 C
 
 ∞
 
 
 (
 X
 )
 
 
 {\displaystyle C^{\infty }(X)}
 
-module of vector fields on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 to the 
 
 
 
 
 C
 
 ∞
 
 
 (
 Y
 )
 
 
 {\displaystyle C^{\infty }(Y)}
 
-module of vector fields on 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
, but this monomorphisms is not a Lie algebra morphism, unless 
 
 
 
 Γ
 
 
 {\displaystyle \Gamma }
 
 is flat.
However, there is a category of above mentioned natural bundles 
 
 
 
 T
 →
 X
 
 
 {\displaystyle T\to X}
 
 which admit the functorial lift 
 
 
 
 
 
 
 τ
 ~
 
 
 
 
 
 {\displaystyle {\widetilde {\tau }}}
 
 onto 
 
 
 
 T
 
 
 {\displaystyle T}
 
 of any vector field 
 
 
 
 τ
 
 
 {\displaystyle \tau }
 
 on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 such that 
 
 
 
 τ
 →
 
 
 
 τ
 ~
 
 
 
 
 
 {\displaystyle \tau \to {\widetilde {\tau }}}
 
 is a Lie algebra monomorphism

[
        
          
            
              τ
              ~
            
          
        
        ,
        
          
            
              
                τ
                ~
              
            
          
          ′
        
        ]
        =
        
          
            
              
                [
                τ
                ,
                
                  τ
                  ′
                
                ]
              
              ~
            
          
        
        .
      
    
    {\displaystyle [{\widetilde {\tau }},{\widetilde {\tau }}']={\widetilde {[\tau ,\tau ']}}.}

This functorial lift 
 
 
 
 
 
 
 τ
 ~
 
 
 
 
 
 {\displaystyle {\widetilde {\tau }}}
 
 is an infinitesimal general covariant transformation of 
 
 
 
 T
 
 
 {\displaystyle T}
 
.
In a general setting, one considers a monomorphism 
 
 
 
 f
 →
 
 
 
 f
 ~
 
 
 
 
 
 {\displaystyle f\to {\widetilde {f}}}
 
 of a group of diffeomorphisms of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 to a group of bundle automorphisms of a natural bundle 
 
 
 
 T
 →
 X
 
 
 {\displaystyle T\to X}
 
. Automorphisms 
 
 
 
 
 
 
 f
 ~
 
 
 
 
 
 {\displaystyle {\widetilde {f}}}
 
 are called the general covariant transformations of 
 
 
 
 T
 
 
 {\displaystyle T}
 
. For instance, no vertical automorphism of 
 
 
 
 T
 
 
 {\displaystyle T}
 
 is a general covariant transformation.
Natural bundles are exemplified by <a href="/facts/Tensor_bundle/cAC6F6rF">tensor bundles</a>. For instance, the <a href="/facts/Tangent_bundle/dq8zW9Kb">tangent bundle</a> 
 
 
 
 T
 X
 
 
 {\displaystyle TX}
 
 of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a natural bundle. Every diffeomorphism 
 
 
 
 f
 
 
 {\displaystyle f}
 
 of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 gives rise to the tangent automorphism 
 
 
 
 
 
 
 f
 ~
 
 
 
 =
 T
 f
 
 
 {\displaystyle {\widetilde {f}}=Tf}
 
 of 
 
 
 
 T
 X
 
 
 {\displaystyle TX}
 
 which is a general covariant transformation of 
 
 
 
 T
 X
 
 
 {\displaystyle TX}
 
. With respect to the holonomic coordinates 
 
 
 
 (
 
 x
 
 λ
 
 
 ,
 
 
 
 
 x
 ˙
 
 
 
 
 λ
 
 
 )
 
 
 {\displaystyle (x^{\lambda },{\dot {x}}^{\lambda })}
 
 on 
 
 
 
 T
 X
 
 
 {\displaystyle TX}
 
, this transformation reads

x
                ˙
              
            
          
          
            ′
            
              μ
            
          
        
        =
        
          
            
              ∂
              
                x
                
                  ′
                  
                    μ
                  
                
              
            
            
              ∂
              
                x
                
                  ν
                
              
            
          
        
        
          
            
              
                x
                ˙
              
            
          
          
            ν
          
        
        .
      
    
    {\displaystyle {\dot {x}}'^{\mu }={\frac {\partial x'^{\mu }}{\partial x^{\nu }}}{\dot {x}}^{\nu }.}

A <a href="/facts/Frame_bundle/Gugt3qyL">frame bundle</a> 
 
 
 
 F
 X
 
 
 {\displaystyle FX}
 
 of linear tangent frames in 
 
 
 
 T
 X
 
 
 {\displaystyle TX}
 
 also is a natural bundle. General covariant transformations constitute a subgroup of holonomic automorphisms of 
 
 
 
 F
 X
 
 
 {\displaystyle FX}
 
. All bundles associated with a frame bundle are natural. However, there are natural bundles which are not associated with 
 
 
 
 F
 X
 
 
 {\displaystyle FX}
 
.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/General_covariance/KzTG2t9g">General covariance</a></li>
<li><a href="/facts/Gauge_gravitation_theory/wwRu94cD">Gauge gravitation theory</a></li>
<li><a href="/facts/Fibered_manifold/4g6exYHA">Fibered manifold</a></li></ul>

<ul><li>Kolář, I., Michor, P., Slovák, J., Natural operations in differential geometry. Springer-Verlag: Berlin Heidelberg, 1993. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-56235-4, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-56235-4.</li>
<li><a href="/facts/Gennadi_Sardanashvily/EDrGkc9r">Sardanashvily, G.</a>, Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory, Lambert Academic Publishing: Saarbrücken, 2013. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-659-37815-7; <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/0908.1886">0908.1886</a></li>
<li>Saunders, D.J. (1989), <a href="https://archive.org/details/geometryofjetbun0000saun">The geometry of jet bundles</a>, Cambridge University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-36948-7</li></ul>

General covariant transformations open-in-new

General covariant transformations