An introduction to Lorentz geometry (Q2889355)

This monograph is devoted to the study of Lorentzian geometry. The authors present the classical approach to the fundamentals but they also add their own research results. All notions related to Lorentzian geometry are introduced in the \(2\)-dimensional Lorentzian plane and then generalizations to the \(n\)-dimensional case follow. The work is divided into six chapters. In Chapter 1, the authors introduce the Lorentzian plane and then all Euclidean analogues. Chapter 2 is devoted to the \(3\)-dimensional Lorentzian space. After introducing a pseudo-sphere and some special surfaces like various cones, the authors consider special cylindrical and stereographical projections. The generalization to \(n\)-dimensional Lorentzian space and special zero-curves whose tangent vector is a zero vector are found in Chapter 3. In Chapter 4, the authors study semi-Riemannian manifolds of signature \((\underbrace {-,\dots,-}_v,\underbrace{+,\dots,+}_{n-v})\). If \(v=0\), then a semi-Riemannian manifold \(M\) is Riemannian and when \(v=1\), then \(M\) is Lorentzian. The Levi-Civita connection and curvatures are introduced. The major part of this chapter is devoted to study Lorentzian manifolds and their submanifolds. Various types of curvatures in a Lorentzian space are discussed in Chapter 5. Generalizations of the Laplace operator in the semi-Riemannian and in particular in Lorentzian manifolds are presented in the last chapter. This monograph is a perfect source for researchers who would like to start their studies in differential geometry or in any other field of mathematics as well.