Prove that curves are geodesics

Question

Suppose $x$ is a coordinate patch such that $g_{11} = 1$ and $g_{12} = 0$. Prove that the $u_1$-curves are geodesics. (Such a patch is called a geodesic coordinate patch)

I don't know how to proceed......

Answer 1

We have ${\rm d}s^2={\rm d}u_1^2+G{\rm d}u_2^2$. You can compute the geodesic equations anyway you like (by using the formula for the $\Gamma_{ij}^k$, Euler-Lagrange equations, or whatever you want) as $$u_1''(t)-G(u_1(t))G'(u_1(t))u_2'(t)^2=0\qquad\mbox{and}\quad u_2''(t)+2\frac{G'(u_1(t))}{G(u_1(t))}u_1'(t)u_2'(t)=0.$$They are satisfied for $u_1(t)=t$ and $u_2(t)={\rm const.}$, so we're done.