How do I calculate 1-forms for a connection?

[Oxymoron]

Why Cant I Calculate 1-forms!?

Ive been given the following formula to find 1-forms:

2\omega_{ab} = e^d i(X_a)i(X_b)\mbox{d}e_d + i(X_b)\mbox{d}e_a - i(X_a)\mbox{d}e_b

and have been asked to find all connection 1-forms. Of course, you can't find these 1-forms without a metric, so here it is:

g = -\mbox{d}t\otimes\mbox{d}t + f(t)^2\hat{g}

where \hat{g} is the metric on some 3-dimensional space of constant curvature c. My orthonormal frame is

e^0 = \mbox{d}t \quad \quad e^i = f(t)\hat{e}^i

where \hat{e}^i is a \hat{g}-orthonormal frame. And note that X_i = \frac{1}{f}\hat{X}_i. Phew! Hope that's not too hard to comprehend.

But I don't know what to put in for d in the index of e. I am aware that you have to sum over d but I am not sure what to sum over. 0,1,2,3? EDIT: Yes, I sum over everything.

 

[Oxymoron]

So here is how I attempted to calculate \omega_{12}:

First I calculated that

\mbox{d}e^1 = f^{\prime}\mbox{d}t\wedge\hat{e}^1 + f\mbox{d}\hat{e}^1
= \frac{f^{\prime}}{f}e^0\wedge e^1 + f\mbox{d}e^1

and, similarly

\mbox{d}e^2 = \frac{f^{\prime}}{f}e^0\wedge e^2 + f\mbox{d}\hat{e}^2

So then, using that equation and substituting d=0 (I don't even know if that's right), I get:

2\omega_{12} = e^di(X_1)i(X_2)\mbox{d}e_d + i(X_2)\mbox{d}e_1 - i(X_1)\mbox{d}e_2

= e^di(X_1)i(X_2)\mbox{d}e_d + i(X_2)\left[\frac{f^{\prime}}{f}e^0\wedge e^1 + f\mbox{d}\hat{e}^1\right] - i(X_1)\left[\frac{f^{\prime}}{f}e^0\wedge e^2 + f\mbox{d}\hat{e}^2\right]

= e^di(X_1)i(X_2)\mbox{d}e_d + i(X_2)(f\mbox{d}\hat{e}^1) - i(X_1)(f\mbox{d}\hat{e}^2) + \frac{f^{\prime}}{f}\left[i(X_2)(e^0\wedge e^1) - i(X_1)(e^0 \wedge e^2)\right]

= e^di(X_1)i(X_2)\mbox{d}e_d + \frac{1}{f}i(\hat{X}_2)(f\mbox{d}\hat{e}^1) - \frac{1}{f}i(\hat{X}_1)(f\mbox{d}\hat{e}^2) + \frac{f^{\prime}}{f}\left[i(X_2)(e^0\wedge e^1) - i(X_1)(e^0 \wedge e^2)\right]

= \frac{1}{f^2}\hat{e}^di(\hat{X}_1)i(\hat{X}_2)\mbox{d}\hat{e}_d + i(\hat{X}_2)(\mbox{d}\hat{e}^1) - i(\hat{X}_1)(\mbox{d}\hat{e}^2) + \frac{f^{\prime}}{f}\left[i(X_2)(e^0\wedge e^1) - i(X_1)(e^0 \wedge e^2)\right]

= \frac{2}{f^2}\hat{\omega}_{12} + \frac{f^{\prime}}{f}\left[i(X_2)(e^0\wedge e^1) - i(X_1)(e^0 \wedge e^2)\right] <br />