Translated from the Portuguese.

<P>1. Welcome to Lorentz-Minkowski Space. 1.1. Pseudo-Euclidean Spaces. 1.2. Subspaces of<STRONG> RQe</STRONG>. 1.3. Contextualization in Special Relativity. 1.4. Isometries in RQe. 1.5. Investigating O1(2, <B>R</B>) And O1(3, <B>R</B>). 1.6 Cross Product in <STRONG>RQe</STRONG>. 2. Local Theory of Curves. 2.1. Parametrized Curves in <STRONG>RQe</STRONG>. 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</P>