The $\mathit{SO}(5)$ Landau model is the mathematical platform of the 4D quantum Hall effect and provide a rare opportunity for a physical realization of the fuzzy four-sphere. We present an integrated analysis of the $\mathit{SO}(5)$ Landau models and the associated matrix geometries through the Landau level projection. With the $\mathit{SO}(5)$ monopole harmonics, we explicitly derive matrix geometry of a four-sphere in any Landau level: In the lowest Landau level the matrix coordinates are given by the generalized $\mathit{SO}(5)$ gamma matrices of the fuzzy four-sphere satisfying the quantum Nambu algebra, while in higher Landau level the matrix geometry becomes a nested fuzzy structure realizing a pure quantum geometry with no counterpart in classical geometry. The internal fuzzy geometry structure is discussed in the view of an $\mathit{SO}(4)$ Pauli-Schrödinger model and the $\mathit{SO}(4)$ Landau model, where we unveil a hidden singular gauge transformation between their background non-Abelian field configurations. Relativistic versions of the $\mathit{SO}(5)$ Landau model are also investigated and relationship to the Berezin-Toeplitz quantization is clarified. We finally discuss the matrix geometry of the Landau models in even higher dimensions.

的 $\mathit{所以}（5）$Landau模型是4D量子霍尔效应的数学平台，为模糊四球的物理实现提供了难得的机会。我们提出了对$\mathit{所以}（5）$通过Landau水平投影，可以找到Landau模型和相关的矩阵几何。随着$\mathit{所以}（5）$ 单极谐波，我们可以明确推导任意Landau层上四球体的矩阵几何：在最低Landau层上，矩阵坐标由广义方程给出 $\mathit{所以}（5）$满足量子Nambu代数的模糊四球体的γ矩阵，而在更高的Landau层次上，矩阵几何成为嵌套的模糊结构，实现了纯量子几何，而经典几何中没有对应的几何。内部模糊几何结构的讨论是从$\mathit{所以}（4）$ Pauli-Schrödinger模型和 $\mathit{所以}（4）$Landau模型，我们在其背景非阿贝尔域配置之间揭示了隐藏的奇异轨距转换。相对论的$\mathit{所以}（5）$还研究了Landau模型，并阐明了与Berezin-Toeplitz量化的关系。最后，我们将讨论甚至更大尺寸的Landau模型的矩阵几何。