Difference between revisions of "Torsion is tensorial"
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Then, <math>\tau(\nabla)</math> is a [[fact about::tensorial map]] in both coordinates. In other words, the value of <math>\tau(\nabla)</math> at a point <math>p \in M</math> depends only on <math>\nabla, X(p), Y(p)</math> and does not depend on the values of the vectors fields <math>X,Y</math> at points other than <math>p</math>.  Then, <math>\tau(\nabla)</math> is a [[fact about::tensorial map]] in both coordinates. In other words, the value of <math>\tau(\nabla)</math> at a point <math>p \in M</math> depends only on <math>\nabla, X(p), Y(p)</math> and does not depend on the values of the vectors fields <math>X,Y</math> at points other than <math>p</math>.  
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+  More explicitly, for any point <math>p \in M</math>, <math>\tau(\nabla)</math> defines a bilinear map:  
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+  <math>\! \tau(\nabla): T_p(M) \times T_p(M) \to T_p(M)</math>  
+  
+  Further, since in fact [[torsion is antisymmetric]], this is an alternating bilinear map.  
==Related facts==  ==Related facts== 
Latest revision as of 17:56, 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
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. In other words, the value of at a point depends only on and does not depend on the values of the vectors fields at points other than .
More explicitly, for any point , defines a bilinear map:
Further, since in fact torsion is antisymmetric, this is an alternating bilinear map.
Related facts
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.