O’Neill Tensor Bounds for Riemannian Submersions in Fibred Calabi–Yau Manifolds

We present a systematic geometric analysis of sectional curvature structures on fibred Calabi-Yau manifolds using the theory of Riemannian submersions and O'Neill's curvature decomposition formulas. Emphasis is placed on elliptic, toroidal, and K3 fibrations arising in complex dimensions one through four. We derive explicit curvature decompositions for horizontal, vertical, and mixed planes and establish quantitative bounds that relate sectional curvature to the tensorial data governing the fibration geometry. These results clarify how rich local curvature phenomena and anisotropies can arise despite the global Ricci-flatness of Calabi-Yau metrics. The framework developed here provides a unified geometric perspective on curvature behavior in fibred Calabi-Yau manifolds and supports both analytical investigations and computational approaches to curvature estimation in explicit geometries.