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In this paper, we prove the existence theorem for longest paths in sub-Lorentzian problems, which generalizes the classical theorem for globally hyperbolic Lorentzian manifolds. We specifically address the case of invariant structures on homogeneous spaces, as the conditions for the existence theorem in this case can be significantly simplified. In particular, it turns out that longest paths exist for any left-invariant sub-Lorentzian structures on Carnot groups.

I review Lorentzian causality theory paying particular attention to the optimality and generality of the presented results. I include complete proofs of some foundational results that are otherwise difficult to find in the literature (e.g. equivalence of some Lorentzian length definitions, upper semi-continuity of the length functional, corner regularization, etc.). The paper is almost self-contained thanks to a systematic logical exposition of the many different topics that compose the theory. It contains new results on classical concepts such as maximizing curves, achronal sets, edges, horismos, domains of dependence, Lorentzian distance. The treatment of causally pathological spacetimes requires the development of some new versatile causality notions, among which I found particularly convenient to introduce: biviability, chronal equivalence, araying sets, and causal versions of horismos and trapped sets. Their usefulness becomes apparent in the treatment of the classical singularity theorems, which is here considerably expanded in the exploration of some variations and alternatives.

This book presents basic General Relativity and provides a basis for understanding and using the fundamental theory. General Relativity is a beautiful geometric theory, simple in its mathematical formulation. It leads to numerous consequences with striking physical interpretations: gravitational waves, black holes, cosmological models, and so on. The first part of the book outlines the fundamentals of the subject. Chapters in this part look at Riemannian and Lorentzian geometry, Special and General Relativity, the Einstein equations, the Schwarzschild spacetime, black holes, and cosmology. The second part presents a number of more advanced topics such as general Einstein spacetimes, the Cauchy problem, relativistic fluids, and Relativistic Kinetic Theory.

We present the authors’ new theory of the RT-equations (‘regularity transformation’ or ‘Reintjes–Temple’ equations), nonlinear elliptic partial differential equations which determine the coordinate transformations which smooth connections Γ to optimal regularity, one derivative smoother than the Riemann curvature tensor Riem(Γ ). As one application we extend Uhlenbeck compactness from Riemannian to Lorentzian geometry; and as another application we establish that regularity singularities at general relativistic shock waves can always be removed by coordinate transformation. This is based on establishing a general multi-dimensional existence theory for the RT-equations by application of elliptic regularity theory in Lp spaces. The theory and results announced in this paper apply to arbitrary L ∞ connections on the tangent bundle T𝓜 of arbitrary manifolds 𝓜, including Lorentzian manifolds of general relativity.

We consider three examples of functors from Lorentzian categories and their applications in finiteness results, singularity theorems and boundary constructions. The third example is a novel functor from the category of ordered measure spaces to the category of Lorentzian pre-length spaces in the sense of Kunzinger–Sämann.

The Epstein-Penner convex hull construction associates to every decorated punctured hyperbolic surface a convex set in the Minkowski space. It works in the de Sitter and anti-de Sitter spaces as well. In these three spaces, the quotient of the spacelike boundary part of the convex set has an induced Euclidean, spherical and hyperbolic metric, respectively, with conical singularities. We show that this gives a bijection from the decorated Teichmüller space to a moduli space of such metrics in the Euclidean and hyperbolic cases, as well as a bijection between specific subspaces of them in the spherical case. Moreover, varying the decoration of a fixed hyperbolic surface corresponds to a discrete conformal change of the metric. This gives a new 3-dimensional interpretation of discrete conformality which is in a sense inverse to the Bobenko-Pinkall-Springborn interpretation.

In this work, we investigate generalized Ricci solitons on four-dimensional non-Abelian Lie groups and undertake a comprehensive classification of left-invariant affine generalized Ricci solitons equipped with left-invariant Lorentzian metrics. Also, we derive that four-dimensional non-Abelian nilpotent Lie groups with left-invariant Lorentzian metric have non-trivial Killing vector fields except for . Furthermore, both and admit non-trivial homogeneous Ricci solitons.

The generalized Schwarzschild spacetimes are introduced as warped manifolds where the base is an open subset of R 2 equipped with a Lorentzian metric and the fiber is a Riemannian manifold. This family includes physically relevant spacetimes closely related to models of black holes. The generalized Schwarzschild spacetimes are endowed with involutive distributions which provide foliations by lightlike hypersurfaces. In this paper, we study spacelike submanifolds immersed in the generalized Schwarzschild spacetimes, mainly, under the assumption that such submanifolds lie in a leaf of the above foliations. In this scenario, we provide an explicit formula for the mean curvature vector field and establish relationships between the extrinsic and intrinsic geometry of the submanifolds. We have derived several characterizations of the slices, and we delve into the specific case where the warping function is the radial coordinate in detail. This subfamily includes the Schwarzschild and Reissner–Nordström spacetimes.

Canonical connections play important roles in studying the differential geometry properties of submanifolds in Lie groups. We define the first kind of canonical connection and the second canonical connection on Lorentzian approximations of the Heisenberg group. Moreover, we give the definitions of intrinsic curvature of a regular curve as well of intrinsic geodesic curvature of regular curves on Lorentzian and spacelike surfaces and of intrinsic Gaussian curvature of Lorentzian and spacelike surfaces away from characteristic points. Furthermore, we derive the expressions of those curvatures and prove Gauss–Bonnet Theorems for the Lorentzian and spacelike surfaces associated to canonical connections in the Lorentzian Heisenberg group.

This study investigates the geometry of osculating type-2 ruled surfaces in Minkowski 3-space E13, formulated through the Type-2 Bishop frame associated with a spacelike curve whose principal normal is timelike and binormal is spacelike. Using the hyperbolic transformation linking the Frenet–Serret and Bishop frames, we analyze how the Bishop curvatures ζ1 and ζ2 affect the geometric behavior and formation of such surfaces. Explicit criteria are derived for cylindrical, developable, and minimal configurations, together with analytical expressions for Gaussian and mean curvatures. We also determine the conditions under which the base curve behaves as a geodesic, asymptotic line, or line of curvature. Several illustrative examples in Minkowski 3-space are provided to visualize the geometric influence of ζ1 and ζ2 on flatness, minimality, and developability. Overall, the Type-2 Bishop frame offers a smooth and effective framework for characterizing Lorentzian geometry and symmetry of osculating ruled surfaces, extending classical Euclidean results to the Minkowski setting.