Torsion is tensorial: Difference between revisions
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given by: | given by: | ||
<math>\tau(\nabla)(X,Y) = \nabla_Y X | <math>\tau(\nabla)(X,Y) = \nabla_X Y - \nabla_Y X - [X,Y]</math> | ||
Then, <math>\tau(\nabla)</math> is a [[tensorial map]] in both coordinates. | Then, <math>\tau(\nabla)</math> is a [[tensorial map]] in both coordinates. |
Revision as of 22:58, 5 April 2008
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 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
- Leibniz rule for derivations: This states that for a vector field and functions , we have:
- Corollary of Leibniz rule for Lie bracket: This states that for a function and vector fields :
- The Leibniz rule axiom that's part of the definition of a connection, namely:
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:
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.
Fill this in later