Pinching on open manifolds

TL;DR

This paper demonstrates that on any open manifold, it is possible to construct incomplete Riemannian metrics with prescribed constant sectional curvature, using the h-principle and jet bundle techniques.

Contribution

It introduces a method to produce arbitrarily pinched Riemannian metrics on open manifolds via the 2-jet bundle and the h-principle, extending previous results to all noncompact manifolds.

Findings

Existence of sections fulfilling curvature relations in jet bundles.

Construction of incomplete metrics with prescribed curvature on open manifolds.

Application of Gromov's h-principle to Riemannian geometry.

Abstract

We show that the 2-jet bundle of local Riemannian metrics on an arbitrary differentiable manifold admits a section which pointwise fulfills the curvature relation sec(g)=a for any real number a. It follows by Gromov's h-principle for open, invariant differential relations that every noncompact differentiable manifold carries arbitrarily pinched (incomplete) Riemannian metrics.