The class of warped product manifolds plays an important role in differential geometry and physics. In this paper, we shall study product manifolds $\mathbb{R} \times M$ with Finsler metrics arising from warped product structure. We give an equivalent condition for those metrics to be of quadratic Weyl curvature.

We prove the existence of quadratic relations between periods of meromorphic flat bundles on complex manifolds with poles along a divisor with normal crossings under the assumption of "goodness". In dimension one, for which goodness is always satisfied, we provide methods to compute the various pairings involved. In an appendix, we give details on the classical results needed for the proofs. V3: Revised version, various proofs simplified in Section 3, exposition improved.

The purpose of the paper is to study the behavior of the classes of functions of bounded lower oscillation under the change of measure given by the weight $w$. More specifically, we provide sharp upper and lower bounds for the norm of the inclusion $BLO\hookrightarrow BLO(w)$ in terms of $A_\infty$ constant of the weight. The results hold in the general context of probability spaces equipped with a treelike structure.