Curvature is tensorial
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 connection on a vector bundle
over a differential manifold
. The Riemann curvature tensor of
is given as a map
defined by:
We claim that is a tensorial map in each of the variables
.
Related facts
- Curvature is antisymmetric in first two variables
- Curvature is antisymmetric in last two variables
- Curvature is symmetric in the pairs of first and last two variables
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. These proofs do not involve any explicit use of
. The proof for
relies simply on repeated application of the product rule, and the fact that
.
Tensoriality in the first variable
Given: is a
-function.
To prove: , or more explicitly,
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 | ![]() |
Fact (1): ![]() ![]() |
![]() |
2 | ![]() |
Fact (2) | ![]() ![]() ![]() ![]() ![]() ![]() |
3 | ![]() |
Fact (1) | ![]() |
4 | ![]() |
![]() |
![]() |
5 | ![]() |
Fact (3) | ![]() |
6 | ![]() |
Fact (1) | ![]() |
Tensoriality in the second variable
Given: is a
-function.
To prove: , or more explicitly,
.
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 | ![]() |
Fact (1) | ![]() |
2 | ![]() |
Fact (2) | ![]() ![]() ![]() ![]() ![]() ![]() |
3 | ![]() |
Fact (1) | ![]() |
4 | ![]() |
![]() |
![]() |
5 | ![]() |
Fact (3) | ![]() |
6 | ![]() |
Fact (1) | ![]() |
Tensoriality in the third variable
Given: A -function
.
To prove: . More explicitly,
.
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 | ![]() |
Fact (2) | ![]() ![]() |
2 | ![]() |
Fact (2) | ![]() |
3 | ![]() |
-- | cancellations |
4 | ![]() |
use ![]() |
cancellation |