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[en] Smooth 2-surfaces with pseudo-Riemannian metric are considered, that is, ones with quadratic form in the tangent bundle that is not positive-definite. Degeneracy points of the form are said to be parabolic. Geodesic lines induced by this pseudo-Riemannian metric in a neighbourhood of typical parabolic points are considered, their phase portraits are obtained and extremal properties are investigated. Bibliography: 23 titles.
[en] We give an exhaustive description of the family of infinite geodesics in the discrete Heisenberg group (with respect to the standard generating set). The classification of infinite geodesics is needed to describe the so-called absolute (exit boundary) of a group. The absolute of the discrete Heisenberg group will be described in a forthcoming paper.
[en] In more than four spacetime dimensions, a multiple Weyl-aligned null direction (WAND) need not be geodesic. It is proved that any higher dimensional Einstein spacetime admitting a non-geodesic multiple WAND also admits a geodesic multiple WAND. All five-dimensional Einstein spacetimes admitting a non-geodesic multiple WAND are determined.
[en] We prove that causal maximizers in C 0,1 spacetimes are either timelike or null. This question was posed in Sämann and Steinbauer (2017 arXiv:1710.10887) since bubbling regions in spacetimes () can produce causal maximizers that contain a segment which is timelike and a segment which is null, see Chruściel and Grant (2012 Class. Quantum Grav. 29 145001). While C 0,1 spacetimes do not produce bubbling regions, the causal character of maximizers for spacetimes with regularity at least C 0,1 but less than C 1,1 was unknown until now. As an application we show that timelike geodesically complete spacetimes are C 0,1-inextendible. (note)
[en] The question considered is whether or not a Riemannian metric can be found to make a given curve field on a closed surface into geodesics. Allowing singularities removes the restriction to Euler characteristic zero. The main results are the following: only two types of isolated singularities can occur in a geodesic field on a surface. No geodsic fields exist on a surface with Euler characteristic less than zero. If the Euler characteristic is zero, such a geodesic field can have only removable singularities. Only a limited number of geodesic fields exist on S2 and RP2. A closed geodesic (perhaps made from several curves and singularities) always appears in such a field
[en] Kim (2008 Class. Quantum Grav. 25 238002) has recently shown how to reconstruct a globally hyperbolic spacetime with non-compact Cauchy surface Σ up to a conformal factor by considering a family of subsets of Σ. We see how this work is related to previous results on reconstructing such spacetimes up to a conformal factor from the set of skies in the space of null geodesics. (comments and replies)
[en] In this paper, we obtain a lower bound for the generalized normalized δ-Casorati curvatures of submanifolds in pointwise Kenmotsu space forms, generalizing two sharp inequalities recently obtained by Lee et al. (Adv. Geom. 2017(3), 355–362, 21) Moreover, we prove that this lower bound is attained at a point p if and only if p is a totally geodesic point. Some examples illustrating the main results of the paper are also given.