AI 中文总结
该研究针对逐点斜黎曼 submersion 的垂直、水平分布建立最优 Casorati 不等式，界定其归一化标量曲率，特殊情形涵盖多种空间形式并给出几何特征，验证了结果的严格性。
AI 中文摘要
我们建立了逐点斜黎曼 submersion 的最优 Casorati 不等式。由于此类 submersion 具有两个不同的分布——垂直分布（切于纤维）和水平分布（由 O'Neill 张量 T 和 A 控制），我们分别对其进行处理。对于从广义复空间形式和广义 Sasakian 空间形式出发的 submersion，我们用各分布的归一化 Casorati 曲率来界定其归一化标量曲率。特殊情形涵盖了实、复、Kähler、Sasakian、Kenmotsu、cosymplectic 以及近、近空间形式的严格不等式。等式具有几何特征：垂直情形下为不变拟脐纤维，水平情形下为可积性。不变与反不变极限重现已知结果，实例说明了严格性。
英文摘要
We establish optimal Casorati inequalities for pointwise slant Riemannian submersions. As such a submersion carries two distinct distributions, the vertical, tangent to the fibers, and the horizontal, governed by O'Neill's tensors T and A, we treat them separately. For submersions from generalized complex and generalized Sasakian space forms we bound the normalized scalar curvature of each distribution by its normalized Casorati curvatures. Special cases recover sharp inequalities for real, complex, Kähler, Sasakian, Kenmotsu, cosymplectic, and almost and almost space forms. Equality is characterized geometrically: invariantly quasi-umbilical fibers in the vertical case and integrability in the horizontal case. The invariant and anti-invariant limits reproduce known results, and examples illustrate sharpness.
