In the Riemannian case, our approach to warped products illuminates curvature formulas that previously seemed formal and somewhat mysterious. Moreover, the geometric approach allows us to study warped products in a much more general class of spaces. For complete metric spaces, it is known that nonpositive curvature in the Alexandrov sense is preserved by gluing on isometric closed convex subsets and by Gromov–Hausdorff limits with strictly positive convexity radius; we show it is also preserved by warped products with convex warping functions.
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