Harmonic Vector Fields and Rigidity of Multiply Warped Products

We investigate harmonic vector fields on multiply warped products in both Riemannian and Lorentzian settings. An explicit decomposition formula for the rough Laplacian acting on vector fields is established, separating horizontal and vertical components and revealing the coupling effects induced by multiple warping functions. This analytic description yields sharp characterizations of harmonicity and leads to strong rigidity phenomena. In particular, we show that, under natural compactness, curvature, and completeness assumptions, the existence of nontrivial harmonic vector fields strongly restricts the warping structure, leading to splitting results and, in the rigid regime, forcing the warping function to obey a specific power‐law expansion. Applications include generalized Robertson–Walker and Kasner‐type spacetimes, where harmonic vector fields impose strong constraints on anisotropic expansion.