Curvature is symmetric in the pairs of first and last two variables
Statement
Suppose is a differential manifold and is a Riemannian metric or pseudo-Riemannian metric on . Suppose is the Levi-Civita connection on and is the Riemann curvature tensor for . (In other words, is the Riemann curvature tensor of the Levi-Civita connection).
We can view as a -tensor as follows:
.
Then, we have:
.
Note that since curvature is antisymmetric in first two variables and curvature is antisymmetric in last two variables, this essentially shows that is a symmetric bilinear form on .
Related facts
- Curvature is tensorial
- Curvature is antisymmetric in first two variables
- Curvature is antisymmetric in last two variables
- First Bianchi identity
- Second Bianchi identity
Facts used
- Curvature is antisymmetric in first two variables
- Curvature is antisymmetric in last two variables
- First Bianchi identity: This states that if is a torsion-free linear connection, and is its Riemann curvature tensor, then:
.
For the corresponding -tensor, we have:
.
Proof
Applying fact (3), we get:
Similar statements, permuting the variables, are:
Consider (1) + (2) - (3) - (4) and uses facts (1) and (2). We get:
This completes the proof.