Solving Derivative Operator Equations: Need Advice

[latentcorpse]

im working through a proof and am stuck on the last line. i can't understand why

$\nabla_a \nabla_b \omega_c - \nabla_b \nabla_a \omega_c + \nabla_b \nabla_c \omega_a - \nabla_c \nabla_b \omega_a + \nabla_c \nabla_a \omega_b - \nabla_a \nabla_c \omega_b=0$?

any advice?

 

[turin]

Is the some reason that $[\nabla_{a},\nabla_{b}]\neq0$ (e.g. are you doing GR, and these are covariant derivatives)?

 

[gabbagabbahey]

Use Wald's definition of the Reimann curvature tensor and appeal to its symmetry properties. (Unless this you are trying to prove the appropriate symmetry property in the first place, in which case you'll want to use 3.1.14 along with the symmetry of the Christoffel connections)

 

[jambaugh]

You haven't given much information but I assume the $$\omega$$ are tangent vectors (co-vectors?) and your identity has to do with the fact that you are using a coordinate basis.

Below the surface of this is the more fundamental Jacobi identity which reflects the underlying associativity of the operator algebra.

Jacobi Identity: $$[[\nabla_a,\nabla_b],\nabla_c] +[[\nabla_b,\nabla_c],\nabla_a]+[[\nabla_c,\nabla_a],\nabla_b]=0$$

Which holds for any algebraic system where the $$\nabla$$ are elements and where the bracket is a commutator of an associative product:
$$[\nabla_a,\nabla_b]\equiv \nabla_a\nabla_b - \nabla_b\nabla_a$$

You will note that if you expand all the commutators in the Jacobi identity all terms will cancel (provided you apply the associativity property of the underlying product).

Now assuming the $$\{\omega_a\}$$ are basis vectors of the tangent space in a coordinate basis i.e. then you can think of them as derivatives of the point function p(x) which maps coordinate values to a point on the manifold. Since the point itself is a scalar (it is by definition invariant under transformations at that point) you can apply either coordinate derivatives $$\partial_a$$ or covariant derivatives equivalently. So taking the l.h.s. of the Jacobi identity and applying it to the "point function" p(x) and you get your identity via:

$$\omega_a \equiv \nabla_a \mathbf{p}(x)$$

I'm a bit fuzzy on whether the coordinate basis condition is necessary. I'll do some more research and check back later. (Anyone else recall?) I know there are issues relating Torsion and the Bianchi identities.