Finding the geodesic function for scalar * function= scalar

metric tensor

In this expression the junk on the left is a scalar. The stuff before the integral is another scalar. The integral is a time-like curve between x₁ and x₂ and at imagine f_gf(x₁) is a lower left corner of the rectangle and f_gf(x₂) is the upper right corner and x₂-_x1 is the length of the base of the rectangle. I know the geodesic is the shortest distance curve paramaerized by this on the condition that the area (the integral) times the scalar₁ = scalar₂. The solution isn't a strait line but could be some curved function. How do I find the function f_gf ?



Do i need to find the 2x2 metric tensor for the 2-d paramaterization curve and solve a pair of differential equations like here ?

http://count.ucsc.edu/~rmont/classes...offelE_Lag.pdf