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>. | ||
+ | |||
+ | ==Related facts== | ||
+ | |||
+ | * [[Curvature is tensorial]] | ||
+ | * [[Torsion is antisymmetric]] | ||
+ | * [[Curvature is antisymmetric in first two variables]] | ||
==Facts used== | ==Facts used== |
Revision as of 17:53, 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
.
Related facts
Facts used
Fact no. | Name | Statement with symbols |
---|---|---|
1 | Any connection is ![]() |
![]() ![]() ![]() ![]() |
2 | The Leibniz-like axiom that is part of the definition of a connection | For a function ![]() ![]() ![]() ![]() |
3 | Corollary of Leibniz rule for Lie bracket (in turn follows from Leibniz rule for derivations | For a function ![]() ![]()
|
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): ![]() |
![]() |
3 | ![]() |
Fact (2): The Leibniz-like 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.