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CURVATURE BOUNDS FOR WARPED PRODUCTS OF METRIC STEPHANIE B. ALEXANDER AND RICHARD L. BISHOP
 

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 1-dimensional base to arbitrary base. We also extend Perelman's CBB dou-
bling theorem [Pm] and Reshetnyak's CBA gluing theorem [R] from 0-dimensional
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 bene ts in the singular setting. For instance, the main construction in An-
cel and Guilbault's paper [AG], proving that the interiors of compact contractible
n-manifolds (n  5) are hyperbolic, when expressed in terms of warped products
becomes an example of Theorem 1.1. Indeed, this theorem answers negatively a

  

Source: Alexander, Stephanie - Department of Mathematics, University of Illinois at Urbana-Champaign

 

Collections: Mathematics