Difference between revisions of "Torsion is tensorial"
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(→Facts used) 

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==Facts used==  ==Facts used==  
−  
−  <math>\  +  { class="sortable" border="1" 
−  +  ! Fact no. !! Name !! Statement with symbols  
−  +    
−  +   1  Any connection is <math>C^\infty</math>linear in its subscript argument  <math>\nabla_{fA} = f\nabla_A</math> for any <math>C^\infty</math>function <math>f</math> and vector field <math>A</math>.  
−  <math>\! f[X,Y] = [fX,Y] + (Yf)X</math>  +   
−  +   2  The Leibnizlike axiom that is part of the definition of a connection  For a function <math>f</math> and vector fields <math>A,B</math>, and a connection <math>\nabla</math>, we have <math>\nabla_A(fB) = (Af)(B) + f\nabla_A(B)</math>  
−  <math>\! f[X,Y] = [X,fY]  (Xf)Y</math>  +   
−  +   3  [[uses::Corollary of Leibniz rule for Lie bracket]] (in turn follows from [[uses::Leibniz rule for derivations]] For a function <math>f</math> and vector fields <math>X,Y</math>:  
−  +  <br><math>\! f[X,Y] = [fX,Y] + (Yf)X</math><br><math>\! f[X,Y] = [X,fY]  (Xf)Y</math>  
−  +  }  
−  
==Proof==  ==Proof== 
Revision as of 17:36, 6 January 2012
This article gives the statement, and possibly proof, that a map constructed in a certain way is tensorial
View other such statements
Contents
Statement
Symbolic statement
Let be a differential manifold and be a linear connection on (viz., is a connection on the tangent bundle of ).
Consider the torsion of , namely:
given by:
Then, is a tensorial map in both coordinates.
Facts used
Fact no.  Name  Statement with symbols 

1  Any connection is linear in its subscript argument  for any function and vector field . 
2  The Leibnizlike axiom that is part of the definition of a connection  For a function and vector fields , and a connection , we have 
3  Corollary of Leibniz rule for Lie bracket (in turn follows from Leibniz rule for derivations  For a function and vector fields :

Proof
Tensoriality in the first coordinate
We'll use the fact that tensoriality is equivalent to linearity.
To prove:
Proof: We prove this by expanding everything out on the left side:
To prove the equality with , we observe that it reduces to showing:
which is exactly what the corollary of Leibniz rule above states.
Tensoriality in the second coordinate
The proof is analogous to that for the first coordinate.
To prove
Proof: We prove this by expanding everything out on the left side:
(the last step uses the corollary of Leibniz rule).
Canceling terms, yields the required result.