Parallelizable manifold

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a differentiable manifold 
 
 
 
 M
 
 
 {\displaystyle M}
 
 of dimension n is called parallelizable if there exist <a href="/facts/Smooth_function/mffL4ch2">smooth</a> <a href="/facts/Vector_field/ccFNuPPp">vector fields</a>

{
        
          V
          
            1
          
        
        ,
        …
        ,
        
          V
          
            n
          
        
        }
      
    
    {\displaystyle \{V_{1},\ldots ,V_{n}\}}

on the manifold, such that at every point 
 
 
 
 p
 
 
 {\displaystyle p}
 
 of 
 
 
 
 M
 
 
 {\displaystyle M}
 
 the <a href="/facts/Tangent_vector/UEq5yffm">tangent vectors</a>

{
        
          V
          
            1
          
        
        (
        p
        )
        ,
        …
        ,
        
          V
          
            n
          
        
        (
        p
        )
        }
      
    
    {\displaystyle \{V_{1}(p),\ldots ,V_{n}(p)\}}

provide a <a href="/facts/Basis_of_a_vector_space/89IPoN6c">basis</a> of the <a href="/facts/Tangent_space/crvpbSeG">tangent space</a> at 
 
 
 
 p
 
 
 {\displaystyle p}
 
. Equivalently, the <a href="/facts/Tangent_bundle/dq8zW9Kb">tangent bundle</a> is a <a href="/facts/Trivial_bundle/SJ7Ek6cI">trivial bundle</a>, so that the associated <a href="/facts/Principal_bundle/bbB2D94B">principal bundle</a> of <a href="/facts/Frame_bundle/Gugt3qyL">linear frames</a> has a global section on 
 
 
 
 M
 .
 
 
 {\displaystyle M.}

A particular choice of such a basis of vector fields on 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is called a <a href="/facts/Parallelization_(mathematics)/YRnxaJuJ">parallelization</a> (or an absolute parallelism) of 
 
 
 
 M
 
 
 {\displaystyle M}
 
.

Parallelizable manifold open-in-new

Parallelizable manifold