In Riemannian geometry, we can introduce a coordinate system over the Riemannian manifold (at least, over a chart), giving n coordinates

xⁱ, i=1,...,n

for an n-dimensional manifold. Locally, at least, this gives a basis for the 1-forms, dxⁱ where d is the exterior derivative. The dual basis for the tangent space is e_i.

Now, let's choose an orthonormal basis for the fibers of T. The rest is index manipulation.

Example

Take a 3-sphere with the radius R and give it polar coordinates α, θ, φ.

e(e_α)/R,

e(e_θ)/R sin(α) and

e(e_φ)/R sin(α) sin(θ)

form an orthonormal basis of T.

Call these e₁, e₂ and e₃. Given the metric η, we can ignore the covariant and contravariant distinction for T.

Then, the dreibein,

e_1=R d\alpha

e_2=R \sin \alpha d\theta

e_3=R \sin \alpha \sin \theta d\phi.

So,

de_1=0

de_2=R \cos \alpha d\alpha \wedge d\theta

de_3=R (\cos \alpha \sin \theta d\alpha \wedge d\phi + \sin \alpha \cos \theta d\theta \wedge d\phi).

from the relation

d_\mathbf{A} e=de+A\wedge e=0,

we get

A_{12}=-\cos \alpha d\theta

A_{13}=-\cos \alpha \sin \theta d\phi

A_{23}=-\cos \theta d\phi.

(d_Aη=0 tells us A is antisymmetric)

So, \mathbf{F}=d\mathbf{A}+\mathbf{A}\wedge \mathbf{A},

F_{12}=\sin\alpha d\alpha\wedge d\theta

F_{13}=\sin \alpha \sin \theta d\alpha\wedge d\phi

F_{23}=\sin^2 \alpha \sin \theta d\theta\wedge d\phi