This chapter develops the basics of differentiable manifolds and semi-Riemannian geometry for the applications in general relativity. It will introduce finitistic substitutes for basic topological notions. We will see that after basic topological notions are available, the basic notions of semi-Riemannian geometry, i.e., vector, tensor, covariant derivative, parallel transportation, geodesic and Riemann curvature, are all essentially finitistic already. Theorems on the existence of spacetime singularities are good examples for analyzing the applicability of infinite and continuous mathematical models to finite physical things. The last section of this chapter will analyze one of Hawking’s singularity theorems, whose common classical proof is non-constructive.
KeywordsTangent Vector Timelike Curve Dual Vector Timelike Geodesic Cauchy Hypersurface
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