Building from ideas of hypercomplex analysis on the quaternionic unit ball, we introduce Hermitian, Riemannian and Kähler-like structures on the latter. These are built from the so-called regular Möbius transformations. Such geometric structures are shown to be natural generalizations of those from the complex setup. Our structures can be considered as more natural, from the hypercomplex viewpoint, than the usual quaternionic hyperbolic geometry. Furthermore, our constructions provide solutions to problems not achieved by hyper-Kähler and quaternion-Kähler geometries when applied to the quaternionic unit ball. We prove that the Riemannian metric obtained in this work yields the same tensor previously computed by Arcozzi–Sarfatti. However, our approach is completely geometric as opposed to the function theoretic methods of Arcozzi–Sarfatti.

基于四元单位球超复分析的思想，我们在后者上引入了厄米特、黎曼和类凯勒结构。它们是根据所谓的常规莫比乌斯变换构建的。这种几何结构被证明是复杂设置中几何结构的自然概括。从超复数的角度来看，我们的结构可以被认为比通常的四元双曲几何更加自然。此外，当应用于四元数单位球时，我们的结构提供了超凯勒和四元数凯勒几何无法实现的问题的解决方案。我们证明，在这项工作中获得的黎曼度量产生了与 Arcozzi-Sarfatti 之前计算的相同的张量。然而，我们的方法完全是几何的，而不是 Arcozzi-Sarfatti 的函数理论方法。