Manifolds and Riemannian Geometry

We present a semi-formal build-up of the mathematical structure of a "manifold," the more abstract generalization of a surface. Then, we discuss notions of Riemannian geometry on a specific class of manifolds, called Riemann manifolds, and discuss generalizations of concepts from calculus to Riemann manifolds. Finally, we end with a discussion of curvature on 3D surfaces and prove Gauss's Theorema Egregium.