| Summary: | Semi-Riemannian manifolds is a subject popular in physics, with applications particularly to modern gravitational theory and electrodynamics. Semi-Riemannian geometry is a branch of differential geometry, similar to Riemannian geometry. In fact, Riemannian geometry is a special case of semi-Riemannian geometry where the scalar product of nonzero vectors is only allowed to be positive. This essay approaches the subject from a mathematical perspective, proving some of the main theorems of semi-Riemannian geometry such as the existence and uniqueness of the covariant derivative of Levi-Civita connection, and some properties of the curvature tensor. Finally, this essay aims to deal with the physical applications of semi-Riemannian geometry. In it, two key theorems are proven - the equivalenceof the Einstein field equations, the foundation of modern gravitational physics, and the Schwarzschild solution to the Einstein field equations. Examples of applications of these theorems are presented.
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