Warped Products Admitting a Curvature Bound

Warped products provide perhaps the major source of examples and counterexamples in metric and Riemannian geometry. Sufficient conditions for a warped product B×f F to have a curvature bound in the sense of Alexandrov, either above or below, are found in [AB 04]. Given the importance of warped products, we want to know if all the known sufficient conditions are needed. Here we prove their necessity. At the time of writing [AB 04], we were optimistic about proving necessity. It turned out that there were several points of difficulty, and adequate tools to handle all of them were not available at the time. For spaces of curvature bounded below, it was necessary to wait for Petrunin’s globalization theorem for incomplete spaces [Pt 12]. Here we use it to make a delicate proof of a gluing theorem on the closure of the subset of the boundary on which the warping function is nonvanishing. For curvature bounded above, we had first to obtain the sharp bound on curvature of subspaces [AB 06]. Here we use it to obtain the correct bound on the fiber. This paper contains a new development of basic properties of warped products of metric spaces, including new properties. Importantly, here we allow nonnegative rather than strictly positive warping functions. The former were introduced in [AB 04], where their treatment was ad hoc and in need of greater precision. These