Unified Connection and Curvature Formulas on Singly, Doubly, and Sequential Warped Product Manifolds

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Abstract

This paper presents a unified differential-geometric treatment of warped product manifolds designed for explicit connection and curvature analysis. After establishing the manifold, tangent, metric, and covariant-derivative framework, the study formalizes Lie-derivative criteria for Killing, 2-Killing, and conformal vector fields and organizes the principal curvature invariants, namely Riemann, sectional, Ricci, and scalar curvature. A complete computation on the round unit sphere validates the framework by recovering the expected Levi-Civita structure, constant sectional curvature 1, Ricci tensor equal to the metric, and scalar curvature 2. Building on this foundation, the manuscript derives systematic Levi-Civita identities, Lie-derivative decompositions, and curvature formulas for singly warped, doubly warped, and sequential warped products. The resulting expressions isolate the roles of base and fibre geometry, Hessian terms, and warping-function gradients in pure and mixed curvature components. These results provide a consistent reference for layered warped geometries and support applications in geometric analysis, Einstein-type equations, and relativity-motivated geometric models