Remarks on global sub-Lorentzian geometry

Marek Grochowski

doi:10.1007/s13324-013-0059-4

Marek Grochowski

Open Access

https://doi.org/10.1007/s13324-013-0059-4

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Abstract

This paper aims at being a starting point for the investigation of the global sub-Lorentzian (or more generally sub-semi-Riemannian) geometry, which is a subject completely not known. We present an adaptation to the fiber bundles setting (and state its elementary proof in the real analytic case) of the decomposition result for indefinite metrics. Using it and the result by Sussmann on global spanning of distributions, we state and prove some global facts on sub-Lorentzian manifolds. We also construct a global control system describing nonspacelike curves and study its controllability properties in case of compact manifolds.

Highlights

The motivation of this note is the lack of global theorems in the sub-Lorentzian or more generally sub-semi-Riemannian geometry
A sub-semi-Riemannian manifold is, by definition, a triplet (M, H, g) where M is a smooth connected and paracompact manifold, H is a smooth bracket generating vector distribution of constant rank on M, rank H < dim M, and g is a semiRiemannian metric on H (H is bracket generating if for every x ∈ M there exists a basis X0, . . . , Xk of H defined near x such that the fields X0, . . . , Xk and all their successive commutators evaluated at x span the whole tangent space Tx M)
As far as the author of this paper is concerned, there is only one known global fact from the general subLorentzian geometry proved in [10], and saying that any two causally related points on a time-oriented globally hyperbolic sub-Lorentzian manifold can be joined by a length maximizing nonspacelike future directed curve

Summary

Introduction

The motivation of this note is the lack of global theorems in the sub-Lorentzian or more generally sub-semi-Riemannian geometry. On certain time-oriented sub-Lorentzian manifolds (for instance on all compact), Theorem 1.2 facilitates this task, namely it allows one to construct a global affine control system having as its trajectories a class of nonspacelike future directed curves. Theorem 1.4 Suppose that (M, H, g) is a compact time-oriented sub-Lorentzian manifold.

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