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

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==Proof==  ==Proof==  
+  To prove tensoriality in a variable, it suffices to show <math>C^\infty</math>linearity in that variable. This is because linearity in <math>C^\infty</math>functions guarantees linearity in a function that is 1 at exactly one point, and zero at others.  
+  
+  The proofs for <math>X</math> and <math>Y</math> are analogous, and rely on manipulation of the Lie bracket <math>[fX,Y]</math> and the property of a connection being <math>C^\infty</math> in the subscript vector.  
===Tensoriality in the first coordinate===  ===Tensoriality in the first coordinate===  
−  +  '''Given''': <math>f:M \to \R</math> is <math>C^\infty</math>function  
−  ''To prove'': <math>\tau(\nabla)(fX,Y) = f\tau(\nabla)(X,Y)</math>  +  '''To prove''': <math>\tau(\nabla)(fX,Y) = f\tau(\nabla)(X,Y)</math> 
−  ''Proof'': We  +  '''Proof''': We start out with the left side: 
−  <math>\tau(\nabla)(fX,Y)  +  <math>\tau(\nabla)(fX,Y)</math> 
−  +  Each step below is obtained from the previous one via some manipulation explained along side.  
−  <math>\  +  { class="sortable" border="1" 
−  +  ! Step no. !! Current status of left side !! Facts/properties used !! Specific rewrites  
−  +    
+   1  <math>\nabla_{fX}(Y)  \nabla_Y(fX)  [fX,Y]</math>  Definition of torsion  whole thing  
+    
+   2  <math>f \nabla_X Y  \nabla_Y(fX)  [fX,Y]</math>  Fact (1): <math>C^\infty</math>linearity of connection in subscript argument  <math>\nabla_{fX} \mapsto f\nabla_X</math>  
+    
+   3  <math>f \nabla_X Y  (f \nabla_Y X + (Yf)(X))  [fX,Y]</math>  Fact (2): The Leibnizlike axiom that's part of the definition of a connection  <math>\nabla_Y(fX) \mapsto f\nabla_YX + (Yf)(X)</math>  
+    
+   4  <math>f \nabla_X Y  f \nabla_Y X  ((Yf)(X) + [fX,Y])</math>  parenthesis rearrangement    
+    
+   5  <math>f \nabla_X Y  f \nabla_Y X  f[X,Y]</math>  Fact (3)  <math>(Yf)(X) + [fX,Y] \mapsto f[X,Y]</math>  
+    
+   6  <math>f(\nabla_X Y  \nabla_Y X  [X,Y])</math>  factor out    
+    
+   7  <math>f\tau(\nabla)(X,Y)</math>  use definition of torsion  <math>\nabla_X Y  \nabla_Y X  [X,Y] \mapsto \tau(\nabla)(X,Y)</math>  
+  }  
===Tensoriality in the second coordinate===  ===Tensoriality in the second coordinate===  
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''To prove'' <math>\tau(\nabla)(X,fY) = f \tau(\nabla)(X,Y)</math>  ''To prove'' <math>\tau(\nabla)(X,fY) = f \tau(\nabla)(X,Y)</math>  
−  ''Proof'':  +  ''Proof'': This is similar to tensoriality in the first coordinate. 
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−  
−  
−  
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Revision as of 17:43, 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
To prove tensoriality in a variable, it suffices to show linearity in that variable. This is because linearity in functions guarantees linearity in a function that is 1 at exactly one point, and zero at others.
The proofs for and are analogous, and rely on manipulation of the Lie bracket and the property of a connection being in the subscript vector.
Tensoriality in the first coordinate
Given: is function
To prove:
Proof: We start out with the left side:
Each step below is obtained from the previous one via some manipulation explained along side.
Step no.  Current status of left side  Facts/properties used  Specific rewrites 

1  Definition of torsion  whole thing  
2  Fact (1): linearity of connection in subscript argument  
3  Fact (2): The Leibnizlike axiom that's part of the definition of a connection  
4  parenthesis rearrangement    
5  Fact (3)  
6  factor out    
7  use definition of torsion 
Tensoriality in the second coordinate
The proof is analogous to that for the first coordinate.
To prove
Proof: This is similar to tensoriality in the first coordinate.