 

Nikos Georgiou and
Brendan Guilfoyle
The causal topology of neutral 4manifolds with null boundary
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Published: 
March 16, 2021. 
Keywords: 
Neutral metric, null boundary, hyperbolic 3space, 3sphere, spaces of constant curvature, geodesic spaces, contact. 
Subject: 
Primary: 53A35; Secondary: 57N13. 


Abstract
This paper considers aspects of 4manifold topology from the point of view of the null cone of a neutral metric, a point of view we call neutral causal topology. In particular, we construct and investigate neutral 4manifolds with null boundaries that arise from canonical 3 and 4dimensional settings.
A null hypersurface is foliated by its normal and, in the neutral case, inherits a pair of totally null planes at each point. This paper focuses on these plane bundles in a number of classical settings.
The first construction is the conformal compactification of flat neutral 4space into the 4ball. The null foliation on the boundary in this case is the Hopf fibration on the 3sphere and the totally null planes in the boundary are integrable. The metric on the 4ball is a conformally flat, scalarflat, positive Ricci curvature neutral metric.
The second constructions are subsets of the 4dimensional space of oriented geodesics in a 3dimensional spaceform, equipped with its canonical neutral metric. We consider all oriented geodesics tangent to a given embedded strictly convex 2sphere. Both totally null planes on this null hypersurface are contact, and we characterize the curves in the null boundary that are Legendrian with respect to either totally null plane bundles. The Reeb vector field associated with the alphaplanes are shown to be the oriented normal lines to geodesics in the surface.
The third is a neutral geometric model for the intersection of two surfaces in a 4manifold. The surfaces are the sets of oriented normal lines to two round spheres in Euclidean 3space, which form Lagrangian surfaces in the 4dimensional space of all oriented lines. The intersection of the boundaries of their normal neighbourhoods form tori that we prove are totally real and Lorentz if the spheres do not intersect.
We conclude with possible topological applications of the three constructions, including neutral Kirby calculus, neutral knot invariants and neutral Casson handles, respectively.


Acknowledgements
N/A


Author information
Nikos Georgiou:
Department of Mathematics
Waterford Institute of Technology
Waterford, Co. Waterford, Ireland
ngeorgiou@wit.ie
Brendan Guilfoyle:
School of Science, Technology, Engineering and Mathematics
Institute of Technology, Tralee
Clash, Tralee, Co. Kerry, Ireland
brendan.guilfoyle@ittralee.ie

