Introduction to Lorentz geometry : curves and surfaces /
Ivo Terek Couto, Alexandre Lymberopoulos.
- 1st.
- 1 online resource : illustrations (black and white)
Translated from the Portuguese.
1. Welcome to Lorentz-Minkowski Space. 1.1. Pseudo-Euclidean Spaces. 1.2. Subspaces of RQe. 1.3. Contextualization in Special Relativity. 1.4. Isometries in RQe. 1.5. Investigating O1(2, R) And O1(3, R). 1.6 Cross Product in RQe. 2. Local Theory of Curves. 2.1. Parametrized Curves in RQe. 2.2. Curves in the Plane. 2.3. Curves in Space. 3. Surfaces in Space. 3.1. Basic Topology of Surfaces. 3.2. Casual type of Surfaces, First Fundamental Form. 3.3. Second Fundamental Form and Curvatures. 3.4. The Diagonalization Problem. 3.5. Curves in Surface. 3.6. Geodesics, Variational Methods and Energy. 3.7. The Fundamental Theorem of Surfaces. 4. Abstract Surfaces and Further Topics. 4.1. Pseudo-Riemannian Metrics. 4.2. Riemann's Classification Theorem. 4.3. Split-Complex Numbers and Critical Surfaces. 4.4 Digression: Completeness and Causality
Lorentz Geometry is a very important intersection between Mathematics and Physics, being the mathematical language of General Relativity. Learning this type of geometry is the first step in properly understanding questions regarding the structure of the universe, such as: What is the shape of the universe? What is a spacetime? What is the relation between gravity and curvature? Why exactly is time treated in a different manner than other spatial dimensions? Introduction to Lorentz Geometry: Curves and Surfaces intends to provide the reader with the minimum mathematical background needed to pursue these very interesting questions, by presenting the classical theory of curves and surfaces in both Euclidean and Lorentzian ambient spaces simultaneously. Features: Over 300 exercises Suitable for senior undergraduates and graduates studying Mathematics and Physics Written in an accessible style without loss of precision or mathematical rigor Solution manual available on www.routledge.com/9780367468644