Difference between revisions of "Torsion is tensorial"

From Diffgeom
Jump to: navigation, search
(Facts used)
(Proof)
Line 31: Line 31:
 
==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===
  
We'll use the fact that tensoriality is equivalent to <math>C^\infty</math>-linearity.
+
'''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 prove this by expanding everything out on the left side:
+
'''Proof''': We start out with the left side:
  
<math>\tau(\nabla)(fX,Y) = \nabla_{fX}(Y) - \nabla_Y(fX) - [fX,Y] = f \nabla_X Y  - f \nabla_Y X - (Yf)(X) - [fX,Y]</math>
+
<math>\tau(\nabla)(fX,Y)</math>
  
To prove the equality with <math>f \tau(\nabla)(X,Y)</math>, we observe that it reduces to showing:
+
Each step below is obtained from the previous one via some manipulation explained along side.
  
<math>\! (Yf)(X) = f[X,Y] - [fX,Y]</math>
+
{| class="sortable" border="1"
 
+
! Step no. !! Current status of left side !! Facts/properties used !! Specific rewrites
which is exactly what the corollary of Leibniz rule above states.
+
|-
 +
| 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 Leibniz-like 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===
Line 53: Line 70:
 
''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'': We prove this by expanding everything out on the left side:
+
''Proof'': This is similar to tensoriality in the first coordinate.
 
 
<math>\tau(\nabla)(X,fY) = \nabla_X(fY) = \nabla_{fY}(X) - [X,fY] = (Xf)(Y) + f \nabla_XY - f\nabla_YX - f[X,Y] - (Xf)Y</math>
 
 
 
(the last step uses the corollary of Leibniz rule).
 
 
 
Canceling terms, yields the required result.
 

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

Statement

Symbolic statement

Let M be a differential manifold and \nabla be a linear connection on M (viz., \nabla is a connection on the tangent bundle TM of M).

Consider the torsion of \nabla, namely:

\tau(\nabla): \Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)

given by:

\tau(\nabla)(X,Y) = \nabla_X Y - \nabla_Y X - [X,Y]

Then, \tau(\nabla) is a tensorial map in both coordinates.

Facts used

Fact no. Name Statement with symbols
1 Any connection is C^\infty-linear in its subscript argument \nabla_{fA} = f\nabla_A for any C^\infty-function f and vector field A.
2 The Leibniz-like axiom that is part of the definition of a connection For a function f and vector fields A,B, and a connection \nabla, we have \nabla_A(fB) = (Af)(B) + f\nabla_A(B)
3 Corollary of Leibniz rule for Lie bracket (in turn follows from Leibniz rule for derivations For a function f and vector fields X,Y:


\! f[X,Y] = [fX,Y] + (Yf)X
\! f[X,Y] = [X,fY] - (Xf)Y

Proof

To prove tensoriality in a variable, it suffices to show C^\infty-linearity in that variable. This is because linearity in C^\infty-functions guarantees linearity in a function that is 1 at exactly one point, and zero at others.

The proofs for X and Y are analogous, and rely on manipulation of the Lie bracket [fX,Y] and the property of a connection being C^\infty in the subscript vector.

Tensoriality in the first coordinate

Given: f:M \to \R is C^\infty-function

To prove: \tau(\nabla)(fX,Y) = f\tau(\nabla)(X,Y)

Proof: We start out with the left side:

\tau(\nabla)(fX,Y)

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 \nabla_{fX}(Y) - \nabla_Y(fX) - [fX,Y] Definition of torsion whole thing
2 f \nabla_X Y  - \nabla_Y(fX) - [fX,Y] Fact (1): C^\infty-linearity of connection in subscript argument \nabla_{fX} \mapsto f\nabla_X
3 f \nabla_X Y  - (f \nabla_Y X + (Yf)(X)) - [fX,Y] Fact (2): The Leibniz-like axiom that's part of the definition of a connection \nabla_Y(fX) \mapsto f\nabla_YX + (Yf)(X)
4 f \nabla_X Y  - f \nabla_Y X - ((Yf)(X) + [fX,Y]) parenthesis rearrangement --
5 f \nabla_X Y  - f \nabla_Y X - f[X,Y] Fact (3) (Yf)(X) + [fX,Y] \mapsto f[X,Y]
6 f(\nabla_X Y - \nabla_Y X - [X,Y]) factor out --
7 f\tau(\nabla)(X,Y) use definition of torsion \nabla_X Y - \nabla_Y X - [X,Y] \mapsto \tau(\nabla)(X,Y)

Tensoriality in the second coordinate

The proof is analogous to that for the first coordinate.

To prove \tau(\nabla)(X,fY) = f \tau(\nabla)(X,Y)

Proof: This is similar to tensoriality in the first coordinate.