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 exercisesSuitable for senior undergraduates and graduates studying Mathematics and Physics Written in an accessible style without loss of precision or mathematical rigorSolution manual available on www.routledge.com/9780367468644
This book intends to provide the reader with the minimum math background needed to pursue interesting questions like what is the relation between gravity and curvature by presenting the classical theory of curves and surfaces in both Euclidean and Lorentzian ambient spaces simultaneously.
1. Welcome to Lorentz-Minkowski Space. 1.1. PseudoEuclidean Spaces.
1.2. Subspaces of R. 1.3. Contextualization in Special Relativity. 1.4.
Isometries in R. 1.5. Investigating O1(2, R) And O1(3, R). 1.6 Cross
Product in R.
2. Local Theory of Curves. 2.1. Parametrized Curves in R.
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. Riemanns Classification
Theorem. 4.3. Split-Complex Numbers and Critical Surfaces. 4.4 Digression:
Completeness and Causality
Ivo Terek Couto, born in Sćo Paulo, graduated with a B.Sc. and a M.Sc. in Mathematics from the Institute of Mathematics and Statistics of the University of Sćo Paulo (IMEUSP). Hes currently pursuing PhD at The Ohio State University in Columbus, Ohio. His study and research interests lie mainly in Differential Geometry and its applications in other areas of Mathematics and Physics, particularly in General Relativity and Classical Mechanics.
Alexandre Lymberopoulos, born in Sćo Paulo, has a PhD in Mathematics from the Institute of Mathematics and Statistics of the University of Sćo Paulo (IMEUSP). He has taught in several higher education institutes in Sćo Paulo and returned to IMEUSP as an Assistant Professor in 2011. His main research interest is in Differential Geometry, particularly in immersions and its interactions with other branches of Science.
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