涉及Riemannian映射和Riemannian子浸入的Casorati不等式最优性及四元数空间形式
Optimal inequalities involving Casorati curvatures along Riemannian maps and Riemannian submersions for quaternionic space form
AI总结：
本文研究了四元数空间形式中Riemannian映射和子浸入的Casorati不等式，揭示了等号情况的几何条件与可积性和准脐性质的关系。
摘要：
本文建立了涉及四元数空间形式的Riemannian映射和Riemannian子浸入的Casorati不等式，并提供了其等号情况的几何特征化。首先，我们推导了Riemannian映射到四元数空间形式的Casorati不等式，并描述了相应的等号情况，显示范围空间的叶是不变的准脐的，且相关的形状算子矩阵是可交换的。接下来，我们获得了涉及基本张量场T和A的Riemannian子浸入从四元数空间形式到Riemannian流形的Casorati不等式，以及其几何解释。特别是，我们证明了与张量场A沿水平分布的等号情况等价于水平分布的可积性。此外，与张量场T沿垂直分布相关的等号情况特征化了纤维，这些纤维是不变的准脐的，且形状算子矩阵是可交换的。最后，同时涉及张量场T和A沿水平和垂直分布的等号情况意味着水平分布的可积性以及纤维的不变准脐性质和相应形状算子的可交换性。
英文摘要：
In this paper, we establish Casorati inequalities for Riemannian maps and Riemannian submersions involving quaternionic space forms, and we provide geometric characterisations of their equality cases. First, we derive Casorati inequalities for Riemannian maps to quaternionic space forms and describe the corresponding equality cases, showing that the leaves of the range space are invariantly quasi-umbilical, and that the associated shape operator matrix commutes. Next, we obtain Casorati inequalities involving the fundamental tensor fields $T$ and $A$ for Riemannian submersions from quaternionic space forms onto Riemannian manifolds, together with their geometric interpretations. In particular, we prove that the equality case corresponding to the tensor field $A$ along the horizontal distribution is equivalent to the integrability of the horizontal distribution. Moreover, the equality case associated with the tensor field $T$ along the vertical distribution characterises fibres that are invariantly quasi-umbilical with a commuting shape operator matrix. Finally, the simultaneous equality cases involving both tensor fields $T$ and $A$ along the horizontal and vertical distributions imply the integrability of the horizontal distribution together with the invariantly quasi-umbilical nature of the fibres and the commutativity of the corresponding shape operators.
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