The metric of the surface a unit sphere is:
$$ds^2 = d\theta ^2 + \cos^2(\theta)d\phi ^2$$
I want to find a locally flat coordinate system $\theta'$ and $\phi'$ at coordinates: $\theta = \pi/4$ and $\phi = 0$
I can do that easily by assumption, but I want to work it in the following way because I am preparing for a spacetime metric in physics.
I started like how Bence Racskó started and did the translation: $$\widetilde{\theta} = \theta - \frac{\pi}{4} $$ then followed his work and reached the following:
$$\delta_{ij}=g'_{ij}(0) = A^k_i A^l_j \widetilde{g}_{ij}(0)$$
Because I want the flat coordinates to be at $\widetilde{\theta}= 0$, then got the following equations:
$$1 = A^1_1A^1_1 + \frac{1}{2} A^2_1 A^2_1$$ $$1 = A^1_2A^1_2 + \frac{1}{2} A^2_2 A^2_2$$ $$0 = A^1_2A^1_1 + \frac{1}{2} A^2_2 A^2_1$$
But those are only 3 equalities for 4 unknowns. How should I continue?
Note that if there is another simpler but also general way I could solve this problem, I do not have a problem. My goal is to be able to transform to locally flat coordinates for any metric.