Hello, I am looking for some nontrivial metric on ℝ^2 invariant under the coordinate transformations defined by the 2x2 matrix [1 a12(θ)] [a21(θ) 1], where aik is some real function of θ. In the same way that the Minkowski metric on ℝ^2 is invariant under Lorentz transformations. Does this metric exist? If not does it exist for some related type of transformations? And why? Are there some other nice features about this kind of transformations/matrices?