A note on warped products

In this note we study the warping functions of some warped products. We give geometric conditions on warped products to obtain simple Riemannian products. For instance, we prove that if the sectional curvature of the warped product M xf N is nonnegative and the Riemannian manifold M is parabolic, then f is constant. We also obtain a sharp curvature integral for the warped product S1 xf N in terms of its Ricci curvature, which gives a characterization of the simple product S1 x N. (c) 2021 Elsevier Inc. All rights reserved.