Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1901.07996

The future is not always open

James D. E. Grant·Michael Kunzinger·Clemens Sämann·Roland Steinbauer
View original source ↗Read source document ↗Export metadata ↓

Abstract

We demonstrate the breakdown of several fundamentals of Lorentzian causality theory in low regularity. Most notably, chronological futures (defined naturally using locally Lipschitz curves) may be non-open, and may differ from the corresponding sets defined via piecewise $C^1$-curves. By refining the notion of a causal bubble from [CG:12],we characterize spacetimes for which such phenomena can occur, and also relate these to the possibility of deforming causal curves of positive length into timelike curves (push-up). The phenomena described here are, in particular, relevant for recent synthetic approaches to low regularity Lorentzian geometry where, in the absence of a differentiable structure, causality has to be based on locally Lipschitz curves.

Explore related subjects

math.DGgr-qcmath-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

James D. E. Grant, Michael Kunzinger, Clemens Sämann, Roland Steinbauer. 2019-09-09. The future is not always open. https://doi.org/10.1007/s11005-019-01213-8

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The signature of geometrically decomposable aspherical 4-manifolds

We construct examples of geometrically decomposable aspherical 4-manifolds with non-zero signature. We show that all such 4-manifolds satisfy the inequality (of Bogomolov--Miyaoka--Yau type) $χ\geq 3|σ|$. We also construct examples attaining the equality that are non-geometric and have non-zero signature. Finally, we prove that for higher graph 4-manifolds, with complex-hyperbolic vertices, the strict inequality always holds. Moreover, we construct infinitely many examples of higher graph 4-manifolds with non-zero signature and prove that the inequality is strict and sharp in this class.

math.DG↗
arXiv2026

Chen-Ricci and Hineva Inequalities For Riemannian Submersions and Riemannian Maps With Applications

The Chen--Ricci inequality provides a sharp upper estimate for the Ricci curvature in terms of the ambient curvature and the squared mean curvature, whereas the Hineva inequality gives a complementary lower estimate. Although Chen--Ricci inequalities have been investigated for Riemannian submersions and Riemannian maps, the corresponding Hineva inequalities have not yet been systematically studied in these settings. In this paper, we fill this gap by establishing sharp Hineva inequalities for Riemannian submersions and Riemannian maps. We also provide alternative and direct proofs of the Chen--Ricci inequalities by working directly with the Ricci curvature, rather than proceeding through scalar-curvature identities and optimization arguments. For Riemannian submersions, we obtain sharp upper and lower estimates along the vertical and mixed distributions, while for Riemannian maps we obtain corresponding estimates along the range distribution. The equality cases are completely characterized, and the simultaneous equality of the Chen--Ricci and Hineva inequalities is investigated. As applications, we obtain the corresponding two-sided Ricci-curvature estimates for Riemannian submersions from real and complex space forms and for Riemannian maps into real and complex space forms. Several examples are presented to demonstrate the sharpness of the obtained inequalities.

math.DG↗
arXiv2026

Extrinsic characterizations of biconservative surfaces in the $4$-dimensional hyperbolic space

Biconservative submanifolds arise as a natural relaxation of the biharmonic condition and play an important role in the submanifold theory. In this paper, we study non-CMC biconservative surfaces with parallel normalized mean curvature vector field (PNMC surfaces) in the four-dimensional hyperbolic space $\mathbb{H}^4$, for which we consider the hyperboloid model. We provide a local extrinsic description of such surfaces, showing that they are generated by a directrix curve lying in a totally geodesic hypersurface $\mathbb{H}^3$ of $\mathbb{H}^4$, through a certain normal flow. This extrinsic classification of non-CMC, PNMC biconservative surfaces in $\mathbb{H}^4$ splits naturally into three cases according to the type of a certain vector field, which can be non-zero null, spacelike or timelike. We also prove that these surfaces are invariant under the action of a parabolic, elliptic, and hyperbolic one-parameter group of isometries of $\mathbb{H}^4$, respectively. Moreover, their full groups of ambient isometries preserving the surfaces are determined. Together with the previous results, the classification of non-CMC, PNMC surfaces in four-dimensional space forms is now complete, from both intrinsic and extrinsic points of view.

math.DG↗