Double vector bundle

<h2 id="definition-and-first-consequences">Definition and first consequences</h2>
<p>A double vector bundle consists of 
  
    
      
        (
        E
        ,
        
          E
          
            H
          
        
        ,
        
          E
          
            V
          
        
        ,
        B
        )
      
    
    {\displaystyle (E,E^{H},E^{V},B)}
  
, where 
</p>
<ol><li>the <i>side bundles</i> 
  
    
      
        
          E
          
            H
          
        
      
    
    {\displaystyle E^{H}}
  
 and 
  
    
      
        
          E
          
            V
          
        
      
    
    {\displaystyle E^{V}}
  
 are vector bundles over the base 
  
    
      
        B
      
    
    {\displaystyle B}
  
,</li>
<li>
  
    
      
        E
      
    
    {\displaystyle E}
  
 is a vector bundle on both side bundles 
  
    
      
        
          E
          
            H
          
        
      
    
    {\displaystyle E^{H}}
  
 and 
  
    
      
        
          E
          
            V
          
        
      
    
    {\displaystyle E^{V}}
  
,</li>
<li>the projection, the addition, the scalar multiplication and the zero map on <i>E</i> for both vector bundle structures are morphisms.</li></ol>
<h2 id="double-vector-bundle-morphism">Double vector bundle morphism</h2>
<p>A double vector bundle morphism 
  
    
      
        (
        
          f
          
            E
          
        
        ,
        
          f
          
            H
          
        
        ,
        
          f
          
            V
          
        
        ,
        
          f
          
            B
          
        
        )
      
    
    {\displaystyle (f_{E},f_{H},f_{V},f_{B})}
  
 consists of maps 
  
    
      
        
          f
          
            E
          
        
        :
        E
        ↦
        
          E
          ′
        
      
    
    {\displaystyle f_{E}:E\mapsto E'}
  
,  
  
    
      
        
          f
          
            H
          
        
        :
        
          E
          
            H
          
        
        ↦
        
          E
          
            H

′
        
      
    
    {\displaystyle f_{H}:E^{H}\mapsto E^{H}{}'}
  
,  
  
    
      
        
          f
          
            V
          
        
        :
        
          E
          
            V
          
        
        ↦
        
          E
          
            V

′
        
      
    
    {\displaystyle f_{V}:E^{V}\mapsto E^{V}{}'}
  
 and  
  
    
      
        
          f
          
            B
          
        
        :
        B
        ↦
        
          B
          ′
        
      
    
    {\displaystyle f_{B}:B\mapsto B'}
  
 such that 
  
    
      
        (
        
          f
          
            E
          
        
        ,
        
          f
          
            V
          
        
        )
      
    
    {\displaystyle (f_{E},f_{V})}
  
 is a bundle morphism from 
  
    
      
        (
        E
        ,
        
          E
          
            V
          
        
        )
      
    
    {\displaystyle (E,E^{V})}
  
 to 
  
    
      
        (
        
          E
          ′
        
        ,
        
          E
          
            V

′
        
        )
      
    
    {\displaystyle (E',E^{V}{}')}
  
, 
  
    
      
        (
        
          f
          
            E
          
        
        ,
        
          f
          
            H
          
        
        )
      
    
    {\displaystyle (f_{E},f_{H})}
  
 is a bundle morphism from 
  
    
      
        (
        E
        ,
        
          E
          
            H
          
        
        )
      
    
    {\displaystyle (E,E^{H})}
  
 to 
  
    
      
        (
        
          E
          ′
        
        ,
        
          E
          
            H

′
        
        )
      
    
    {\displaystyle (E',E^{H}{}')}
  
, 
  
    
      
        (
        
          f
          
            V
          
        
        ,
        
          f
          
            B
          
        
        )
      
    
    {\displaystyle (f_{V},f_{B})}
  
 is a bundle morphism from 
  
    
      
        (
        
          E
          
            V
          
        
        ,
        B
        )
      
    
    {\displaystyle (E^{V},B)}
  
 to 
  
    
      
        (
        
          E
          
            V

′
        
        ,
        
          B
          ′
        
        )
      
    
    {\displaystyle (E^{V}{}',B')}
  
 and 
  
    
      
        (
        
          f
          
            H
          
        
        ,
        
          f
          
            B
          
        
        )
      
    
    {\displaystyle (f_{H},f_{B})}
  
 is a bundle morphism from 
  
    
      
        (
        
          E
          
            H
          
        
        ,
        B
        )
      
    
    {\displaystyle (E^{H},B)}
  
 to 
  
    
      
        (
        
          E
          
            H

′
        
        ,
        
          B
          ′
        
        )
      
    
    {\displaystyle (E^{H}{}',B')}
  
.
</p><p>The '<i>flip</i> of the double vector bundle 
  
    
      
        (
        E
        ,
        
          E
          
            H
          
        
        ,
        
          E
          
            V
          
        
        ,
        B
        )
      
    
    {\displaystyle (E,E^{H},E^{V},B)}
  
 is the double vector bundle 
  
    
      
        (
        E
        ,
        
          E
          
            V
          
        
        ,
        
          E
          
            H
          
        
        ,
        B
        )
      
    
    {\displaystyle (E,E^{V},E^{H},B)}
  
.
</p>
<h2 id="examples">Examples</h2>
<p>If 
  
    
      
        (
        E
        ,
        M
        )
      
    
    {\displaystyle (E,M)}
  
 is a vector bundle over a differentiable manifold 
  
    
      
        M
      
    
    {\displaystyle M}
  
 then 
  
    
      
        (
        T
        E
        ,
        E
        ,
        T
        M
        ,
        M
        )
      
    
    {\displaystyle (TE,E,TM,M)}
  
 is a double vector bundle when considering its <a href="/facts/Secondary_vector_bundle_structure/YWT6XE5k">secondary vector bundle structure</a>.
</p><p>If 
  
    
      
        M
      
    
    {\displaystyle M}
  
 is a differentiable manifold, then its <a href="/facts/Double_tangent_bundle/TEMAa6mR">double tangent bundle</a> 
  
    
      
        (
        T
        T
        M
        ,
        T
        M
        ,
        T
        M
        ,
        M
        )
      
    
    {\displaystyle (TTM,TM,TM,M)}
  
 is a double vector bundle.
</p>

<p>Mackenzie, K. (1992), "Double Lie algebroids and second-order geometry, I", <i><a href="/facts/Advances_in_Mathematics/kXtcl4pM">Advances in Mathematics</a></i>, 94 (2): 180–239, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0001-8708%2892%2990036-k">10.1016/0001-8708(92)90036-k</a>
</p>

Double vector bundle open-in-new

Double vector bundle