In General Relativity spacetime is described mathematically by a Lorentzian manifold. Gravitation manifests itself as the curvature of this manifold. Physical fields, such as the electromagnetic field, are defined on this manifold and have to satisfy a wave equation. This book provides an introduction to the theory of linear wave equations on Lorentzian manifolds. In contrast to other texts on this topic [Friedlander1975, Günther1988] we develop the global theory. This means, we ask for existence and uniqueness of solutions which are defined on all of the underlying manifold. Such results are of great importance and are already used much in the literature despite the fact that published proofs are missing. Tracing back the references one typically ends at Leray's unpublished lecture notes [Leray1953] or their exposition [Choquet-Bruhat1968].In this text we develop the global theory from scratch in a modern geometric language. In the first chapter we provide basic definitions and facts about distributions on manifolds, Lorentzian geometry, and normally hyperbolic operators. We study the building blocks for local solutions, the Riesz distributions, in some detail. In the second chapter we show how to solve wave equations locally. Using Riesz distributions and a formal recursive procedure one first constructs formal fundamental solutions. These are formal series solving the equations formally but in general they do not converge. Using suitable cut-offs one gets "almost solutions" from these formal solutions. They are well-defined distributions but solve the equation only up to an error term. This is then corrected by some further analysis which yields true local fundamental solutions.This procedure is similar to the construction of the heat kernel for a Laplace type operator on a compact Riemannian manifold. The analogy goes even further. Similar to the short-time asymptotics for the heat kernel, the formal fundamental solution turns out to be an asymptotic expansion of the true fundamental solution. Along the diagonal the coefficients of this asymptotic expansion are given by the same algebraic expression in the curvature of the manifold, the coefficients of the operator, and their derivatives as the heat kernel coefficients.In the third chapter we use the local theory to study global solutions. This means we construct global fundamental solutions, Green's operators, and solutions to the Cauchy problem. This requires assumptions on the geometry of the underlying manifold. In Lorentzian geometry one has to deal with the problem that there is no good analog for the notion of completeness of Riemannian manifolds. In our context globally hyperbolic manifolds turn out to be the right class of manifolds to consider. Most basic models in General Relativity turn out to be globally hyperbolic but there are exceptions such as iv anti-deSitter spacetime. This is why we also include a section in which we study cases where one can guarantee existence (but not uniqueness) of global solutions on certain non-globally hyperbolic...

We give a description of all complete simply connected Riemannian manifolds carrying real Killing spinors. Furthermore, we present a construction method for manifolds with the exceptional holonomy groups G 2 and Spin (7).

Green-hyperbolic operators are linear differential operators acting on sections of a vector bundle over a Lorentzian manifold which possess advanced and retarded Green's operators. The most prominent examples are wave operators and Dirac-type operators. This paper is devoted to a systematic study of this class of differential operators. For instance, we show that this class is closed under taking restrictions to suitable subregions of the manifold, under composition, under taking "square roots", and under the direct sum construction. Symmetric hyperbolic systems are studied in detail.2010 Mathematics Subject Classification. 58J45,35L45,35L51,35L55,81T20. Key words and phrases. Globally hyperbolic Lorentzian manifolds, Green-hyperbolic operators, wave operators, normally hyperbolic operators, Dirac-type operators, Green's operators, support system, symmetric hyperbolic system, Cauchy problem, energy estimate, finite propagation speed, locally covariant quantum field theory. 1 1.1. Cauchy hypersurfaces. A subset Σ ⊂ M is called a Cauchy hypersurface if every inextensible timelike curve in M meets Σ exactly once. Any Cauchy hypersurface is a topological submanifold of codimension 1. All Cauchy hypersurfaces of M are homeomorphic. If M possesses a Cauchy hypersurface then M is called globally hyperbolic. This class of Lorentzian manifolds contains many important examples: Minkowski space, Friedmann models, the Schwarzschild model and deSitter spacetime are globally hyperbolic. Bernal and Sánchez proved an important structural result [6, Thm. 1.1]: Any globally hyperbolic Lorentzian manifold has a Cauchy temporal function. This is a smooth function t : M → R with past-directed timelike gradient ∇t such that the levels t −1 (s) are (smooth spacelike) Cauchy hypersurfaces if nonempty.1.2. Future and past. From now on let M always be globally hyperbolic. For any x ∈ M we denote by J + (x) the set all points that can be reached by future-directed causal curves emanating from x. For any subset A ⊂ M we put J + (A) := x∈A J + (x). If A is closed so is J + (A). We call a subset A ⊂ M strictly past compact if it is closed and there is aWe denote by I + (x) the set of all points in M that can be reached by future-directed timelike curves emanating from x.Interchanging the roles of future and past, we similarly define J − (x), J − (A), I − (x), strictly future compact and past compact subsets of M. If A ⊂ M is past compact and future compact then we call A temporally compact. For any compact subsetscompact. Similarly, strictly future compact sets are future compact. If we want to emphasize the ambient manifold M, then we write J + M (x) instead of J + (x) and similarly for J − M (x), J ± M (A), and I ± M (A). GREEN-HYPERBOLIC OPERATORS ON GLOBALLY HYPERBOLIC SPACETIMES 3 Example 1.1. Let M be Minkowski space and let C ⊂ M be an open cone with tip 0 containing the closed cone J − (0) \ {0}. Then A = M \ C is past compact but not strictly past compact. Indeed, for each x ∈ M, the set J −is not strictly past compact because the in...

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