This chapter introduces the fundamentals of differentiable manifolds and semi-Riemannian geometry, essential for applications in general relativity. It focuses on developing finitistic substitutes for basic topological concepts, demonstrating that key geometric notions such as vectors, tensors, covariant derivatives, parallel transportation, geodesics, and Riemann curvature are essentially finitistic. The chapter also examines the applicability of infinite and continuous mathematical models to finite physical phenomena, using theorems on spacetime singularities as examples. Specifically, it analyzes Hawking's singularity theorem, showing that its proof can be transformed into valid logical deductions about real spacetime, even if spacetime is discrete at the microscopic scale. The chapter follows classical presentations by Wald, O'Neill, and Naber, focusing on the mathematical aspects while ignoring physical details. In the section on differentiable manifolds, the chapter defines manifolds in strict finitism by replacing non-finitistic topological notions with finitistic ones. It introduces rational open balls and regular open subsets of \(\mathbb{R}^n\), which are uniquely determined by sequences of rational numbers. Regular open sets are open sets in the sense defined in the previous chapter, but only regular open subsets are considered due to their uniform representation and quantification. The chapter also discusses the well-contained property of balls within regular open sets and the continuity of functions on regular open sets.This chapter introduces the fundamentals of differentiable manifolds and semi-Riemannian geometry, essential for applications in general relativity. It focuses on developing finitistic substitutes for basic topological concepts, demonstrating that key geometric notions such as vectors, tensors, covariant derivatives, parallel transportation, geodesics, and Riemann curvature are essentially finitistic. The chapter also examines the applicability of infinite and continuous mathematical models to finite physical phenomena, using theorems on spacetime singularities as examples. Specifically, it analyzes Hawking's singularity theorem, showing that its proof can be transformed into valid logical deductions about real spacetime, even if spacetime is discrete at the microscopic scale. The chapter follows classical presentations by Wald, O'Neill, and Naber, focusing on the mathematical aspects while ignoring physical details. In the section on differentiable manifolds, the chapter defines manifolds in strict finitism by replacing non-finitistic topological notions with finitistic ones. It introduces rational open balls and regular open subsets of \(\mathbb{R}^n\), which are uniquely determined by sequences of rational numbers. Regular open sets are open sets in the sense defined in the previous chapter, but only regular open subsets are considered due to their uniform representation and quantification. The chapter also discusses the well-contained property of balls within regular open sets and the continuity of functions on regular open sets.