# Gromov, Cauchy and Causal Boundaries for Riemannian, Finslerian and Lorentzian Manifolds

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*J. L. Flores; J. Herrera; M. Sánchez*

Recently, the old notion of causal boundary for a
spacetime \(V\) has been redefined consistently. The
computation of this boundary \(\partial V\) on any standard
conformally stationary spacetime \(V=\mathbb{R}\times M\),
suggests a natural compactification \(M_B\) associated to any
Riemannian metric on \(M\) or, more generally, to any
Finslerian one. The corresponding boundary \(\partial_BM\) is
constructed in terms of Busemann-type functions. Roughly,
\(\partial_BM\) represents the set of all the directions in
\(M\) including both, asymptotic and “finite” (or
“incomplete”) directions.

This Busemann boundary
\(\partial_BM\) is related to two classical boundaries: the
Cauchy boundary \(\partial_{C}M\) and the Gromov boundary
\(\partial_GM\).

The authors' aims are: (1) to study the
subtleties of both, the Cauchy boundary for any generalized
(possibly non-symmetric) distance and the Gromov
compactification for any (possibly incomplete) Finsler manifold, (2)
to introduce the new Busemann compactification \(M_B\),
relating it with the previous two completions, and (3) to give a full
description of the causal boundary \(\partial V\) of any
standard conformally stationary spacetime.

#### Table of Contents

# Table of Contents

## Gromov, Cauchy and Causal Boundaries for Riemannian, Finslerian and Lorentzian Manifolds

- Chapter 1. Introduction 18 free
- Chapter 2. Preliminaries 916 free
- Chapter 3. Cauchy completion of a generalized metric space 1522
- Chapter 4. Riemannian Gromov and Busemann completions 2936
- Chapter 5. Finslerian completions 3946
- Chapter 6. C-boundary of standard stationary spacetimes 5966
- 6.1. Chronological relations and Lipschitzian functions 5966
- 6.2. Future and past c-boundaries as point sets 6067
- 6.3. The (total) c-boundary as a point set 6269
- 6.4. Causality of the c-boundary 6673
- 6.5. Topology of the partial boundaries and the c-boundary 6875
- 6.6. Proof of Theorem 1.2 7077
- Acknowledgments 7077

- Bibliography 7582