From CT scans to 4-manifold topology

TL;DR

This survey explores the ultrahyperbolic equation in four dimensions, linking geometric structures, analytic solutions, and topological generalizations of 4-manifolds, providing new insights and solutions in the field.

Contribution

It offers a comprehensive overview of the ultrahyperbolic equation in 4D, introducing a new solution to the Cauchy problem and connecting geometric, analytic, and topological perspectives.

Findings

Introduces a new solution to the Cauchy problem for the ultrahyperbolic equation.

Establishes connections between geometry, analysis, and topology in 4-manifold theory.

Provides a survey of recent developments in the field.

Abstract

In this survey paper the ultrahyperbolic equation in dimension four is discussed from a geometric, analytic and topological point of view. The geometry centres on the canonical neutral metric on the space of oriented geodesics of 3-dimensional space-forms, the analysis discusses a mean value theorem for solutions of the equation and presents a new solution of the Cauchy problem over a certain family of null hypersurfaces, while the topology relates to generalizations of codimension two foliations of 4-manifolds.