First Bianchi identity
From Diffgeom
Contents
Statement
Let be a torsion-free linear connection. The Riemann curvature tensor
of
satisfies the following first Bianchi identity or algebraic Bianchi identity:
for any three vector fields .
Notice that since this proof is applicable for any torsion-free linear connection, it in particular holds for the Levi-Civita connection arising from a Riemannian metric or pseudo-Riemannian metric.
Related facts
- Second Bianchi identity (also called the differential Bianchi identity).
- Curvature is tensorial
- Torsion is tensorial
Proof
Using repeated simplication and the Jacobi identity
Let us plug the definition of the Riemann curvature tensor:
This can be regrouped as:
Now, since is torsion-free, we have
and similar simplifications yield:
again using the fact that is torsion-free, this simplifies to:
this becomes zero by the Jacobi identity.
Using the differential Bianchi identity
Fill this in later