 
Summary: CURVATURE BOUNDS FOR WARPED PRODUCTS OF METRIC
SPACES
STEPHANIE B. ALEXANDER AND RICHARD L. BISHOP
1. Introduction
1.1. Main theorems. This paper gives sharp conditions for a warped product of
metric spaces to have a given curvature bound in the sense of Alexandrov. Thus we
have a broad new construction of spaces with curvature bounds, either above (CBA)
or below (CBB). As applications, we extend the standard cone and suspension con
structions of spaces with curvature bounds [Be, BGP], introduced by Berestovskii,
from 1dimensional base to arbitrary base. We also extend Perelman's CBB dou
bling theorem [Pm] and Reshetnyak's CBA gluing theorem [R] from 0dimensional
ber to arbitrary ber. The proof studies the analytic behavior of billiard trajecto
ries that approximate warped product geodesics, and the geometry of generalized
cone points (vanishing points of the warping function).
In Riemannian geometry, warped products are an important source of construc
tions and counterexamples under curvature constraints [Ps], and we expect even
more benets in the singular setting. For instance, the main construction in An
cel and Guilbault's paper [AG], proving that the interiors of compact contractible
nmanifolds (n 5) are hyperbolic, when expressed in terms of warped products
becomes an example of Theorem 1.1. Indeed, this theorem answers negatively a
