Ivo Terek Couto, born in São Paulo, graduated with a [...]. and a [...]. in Mathematics from the Institute of Mathematics and Statistics of the University of São Paulo (IME-USP). He's 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 São Paulo, has a PhD in Mathematics from the Institute of Mathematics and Statistics of the University of São Paulo (IME-USP). He has taught in several higher education institutes in São Paulo and returned to IME-USP 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.

1. Welcome to Lorentz-Minkowski Space. 1.1. Pseudo-Euclidean 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. Riemann's Classification Theorem. 4.3. Split-Complex Numbers and Critical Surfaces. 4.4 Digression: Completeness and Causality