Dependence does not mean correlation. Two random variables X and Y can be dependent in a non linear way (for instance, X is Beta and Y is X squared) so that their correlation coefficient is zero.
Can exist two gaussians X and Y which are dependent but with zero linear correlation coefficient?
Dependence between gaussians is everywhere modeled as linear e.g. one variable is $(\mu_x + \sigma_x X)$ and the other one is $(Y = \mu_y + \sigma_y(\rho X + \sqrt{1-\rho^2}))$.
But is there a theorem which says this is the only type of dependence between gaussians that exists? Or just for simplicity is it the most utilized?