Warped products and synthetic lower curvature bounds: an overview
Christian Ketterer
TL;DR
This survey unifies warped product constructions across smooth and non-smooth contexts, linking semi-Riemannian, metric, and Lorentzian geometry. It develops geodesic and curvature formulas, establishes fiber independence and energy relations, and derives synthetic curvature bounds (CD, MCP, RCD) for warped products with weighted measures. The work highlights how base curvature, fiber structure, and warping function interact to preserve or propagate sectional, Ricci, and Bakry–Emery curvature bounds in both Riemannian and Lorentzian settings, including Alexandrov and Lorentzian length-space frameworks. By interpreting classical cones, suspensions, and generalized cones as warped products, the article provides a broad toolkit for constructing and analyzing spaces with lower curvature bounds in smooth and non-smooth worlds, with explicit results and conjectures for synthetic curvature-dimension bounds in warped-product geometries.
Abstract
This is a survey about the contruction of warped products between (semi-)Riemannian manifolds and metric (measure) spaces. The resulting spaces will be semi-Riemannian manifolds, metric (measure) spaces or Lorentzian metric and metric measure spaces. We present details of the contruction in each case and we will highlight important properties like fiber independence and the energy equation. Warped products behave nicely in relation with curvature lower bounds. Here we will focus on sectional and Ricci curvature lower bounds and their Lorentzian counterparts. Throughout the article we provide many examples and formulate questions and conjectures.