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Prove that, in semigeodesic coordinates $(x, y)$, every curve of the form $y = const$ is a geodesic.

So, I know from the book this is from the definition that states coordinates $(u^1, u^2) = (x, y)$ are called semigeodesic if the first fundamental form has the shape

$g_{ij}du^idu^j = dx^2 + Gdy^2$

And I also know a geodesic means that at every point of the curve $γ(t) $ the acceleration $γ¨(t)$ is either
zero or parallel to its unit normal $hat n$. I'm assuming I may need to define the coordinates in $Bbb R^3$ by formulas and go from there, however I'm not sure, I'm mainly going off of the example given to us in the book.

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Prove that the Gaussian curvature of a surface with semigeodesic coordinates (x, y) is equal to

$K = -left(frac{1}{sqrt G}right) left(frac{partial^2sqrt G}{partial x^2}right)$

To do this, I feel as though I can use the Gaussian curvature formula with Christoffel symbols that simplifies to
$K = -left(frac{1}{2sqrt {EG}}right)left[left(frac{E_v}{sqrt {EG}}right)_v + left(frac{G_u}{sqrt {EG}}right)_uright]$ (solution to 4.3 problem 1 of Do Carmo), but I'm not certain.

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Prove that if Gaussian curvature K of a surface is constant and $K ne 0$, then there exists semigeodesic coordinates $(x, y)$ in which the first fundamental form has the shape

$begin{pmatrix} 1 & 0 \ 0 & sin^2(sqrt K)xend{pmatrix} for K > 0$

$begin{pmatrix} 1 & 0 \ 0 & sinh^2(sqrt K)xend{pmatrix} for K < 0$

Not sure where to start on this one, unfortunately, so any point in the right direction would be helpful.

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Just in general I'm not very strong with semigeodesic concept so I am struggling with this problem set.