This proceedings volume gathers selected, revised papers presented at the X International Meeting on Lorentzian Geometry (GeLoCor 2021), virtually held at the University of Córdoba, Spain, on February 1-5, 2021. It includes surveys describing the state-of-the-art in specific areas, and a selection of the most relevant results presented at the conference. Taken together, the papers offer an invaluable introduction to key topics discussed at the conference and an overview of the main techniques in use today. This volume also gathers extended revisions of key studies in this field. Bringing new results and examples, these unique contributions offer new perspectives to the original problems and, in most cases, extend and reinforce the robustness of previous findings. Hosted every two years since 2001, the International Meeting on Lorentzian Geometry has become one of the main events bringing together the leading experts on Lorentzian geometry. In this volume, the reader will find studies on spatial and null hypersurfaces, low regularity in general relativity, conformal structures, Lorentz-Finsler spacetimes, and more. Given its scope, the book will be of interest to both young and experienced mathematicians and physicists whose research involves general relativity and semi-Riemannian geometry.

Offers insight into the methods and concepts of a very active field of mathematics that has many connections with physics. It includes contributions from specialists in differential geometry and mathematical physics, collectively demonstrating the wide range of applications of Lorentzian geometry, and ranging in character from research papers to surveys to the development of new ideas.

Traditionally, Lorentzian geometry has been used as a necessary tool to understand general relativity, as well as to explore new genuine geometric behaviors, far from classical Riemannian techniques. Recent progress has attracted a renewed interest in this theory for many researchers: long-standing global open problems have been solved, outstanding Lorentzian spaces and groups have been classified, new applications to mathematical relativity and high energy physics have been found, and further connections with other geometries have been developed. Samples of these fresh trends are presented in this volume, based on contributions from the VI International Meeting on Lorentzian Geometry, held at the University of Granada, Spain, in September, 2011. Topics such as geodesics, maximal, trapped and constant mean curvature submanifolds, classifications of manifolds with relevant symmetries, relations between Lorentzian and Finslerian geometries, and applications to mathematical physics are included. ​ This book will be suitable for a broad audience of differential geometers, mathematical physicists and relativists, and researchers in the field.

This volume consists of expanded versions of invited lectures given at The Beemfest: Advances in Differential Geometry and General Relativity (University of Missouri-Columbia) on the occasion of Professor John K. Beem's retirement. The articles address problems in differential geometry in general and in particular, global Lorentzian geometry, Finsler geometry, causal boundaries, Penrose's cosmic censorship hypothesis, the geometry of differential operators with variable coefficients on manifolds, and asymptotically de Sitter spacetimes satisfying Einstein's equations with positive cosmological constant. The book is suitable for graduate students and research mathematicians interested in differential geometry.

This volume covers material presented by invited speakers at the AMS special session on Riemannian and Lorentzian geometries held at the annual Joint Mathematics Meetings in Baltimore. Topics covered include classification of curvature-related operators, curvature-homogeneous Einstein 4-manifolds, linear stability/instability singularity and hyperbolic operators of spacetimes, spectral geometry of holomorphic manifolds, cut loci of nilpotent Lie groups, conformal geometry of almost Hermitian manifolds, and also submanifolds of complex and contact spaces. This volume can serve as a good reference source and provide indications for further research. It is suitable for graduate students and research mathematicians interested in differential geometry.

This book provides an introduction to and survey of recent developments in pseudo-Riemannian geometry, including applications in mathematical physics, by leading experts in the field. Topics covered are: Classification of pseudo-Riemannian symmetric spaces Holonomy groups of Lorentzian and pseudo-Riemannian manifolds Hypersymplectic manifolds Anti-self-dual conformal structures in neutral signature and integrable systems Neutral Kahler surfaces and geometric optics Geometry and dynamics of the Einstein universe Essential conformal structures and conformal transformations in pseudo-Riemannian geometry The causal hierarchy of spacetimes Geodesics in pseudo-Riemannian manifolds Lorentzian symmetric spaces in supergravity Generalized geometries in supergravity Einstein metrics with Killing leaves The book is addressed to advanced students as well as to researchers in differential geometry, global analysis, general relativity and string theory. It shows essential differences between the geometry on manifolds with positive definite metrics and on those with indefinite metrics, and highlights the interesting new geometric phenomena, which naturally arise in the indefinite metric case. The reader finds a description of the present state of the art in the field as well as open problems, which can stimulate further research.