We investigate mappings $F = (f_1, f_2) \colon \mathbb{R}^2 \to \mathbb{R}^2 $ where $ f_1, f_2 $ are bivariate normal densities from the perspective of singularity theory of mappings, motivated by the need to understand properties of two-component bivariate normal mixtures. We show a classification of mappings $ F = (f_1, f_2) $ via $\mathcal{A}$-equivalence and characterize them using statistical notions. Our analysis reveals three distinct types, each with specific geometric properties. Furthermore, we determine the upper bounds for the number of modes in the mixture for each type.

翻译：我们从映射奇点理论的视角研究映射 $F = (f_1, f_2) \colon \mathbb{R}^2 \to \mathbb{R}^2$，其中 $f_1, f_2$ 为二元正态密度函数，研究动机源于理解双分量二元正态混合分布性质的需求。我们通过 $\mathcal{A}$-等价性对映射 $F = (f_1, f_2)$ 进行分类，并运用统计学概念对其进行表征。分析揭示了三种具有特定几何性质的不同类型。此外，我们确定了每种类型混合分布中模态数量的上界。