Geodesic Equation - Physics Explained

[Euphemia]

http://mykomica.org/boards/shieiuping/physics/src/1335180831965.jpg
http://mykomica.org/boards/shieiuping/physics/src/1335180965708.jpg

 

[Euphemia]

The bigger words is the main description.
The smaller words is my own work,
I don't know if I get the wrong Christoffel connections or something else.

 

[lavinia]

The Lie bracket of the two unit vector fields, (1 + z$\bar{z}$)$\partial$x

and (1 + z$\bar{z}$)$\partial$y

is orthogonal to (1 + z$\bar{z}$)$\partial$x
along the x-axis and so is tangent to a geodesic.

Now use the symmetry of the metric to solve the general case.