A Brief Survey Of Manifolds And Vector Bundles

This is a survey of various key components and machinery used in modern differential geometry. We begin with an introduction to multilinear algebra up to basic notions of tensors and exterior algebras. A survey of basic topology and manifold theory is then presented. This includes a formulation of tangent spaces, vector bundles, and sections on a manifold. This paper then concludes with an introduction to semi-Riemannian geometry including an outline of metrics, connections, and covariant differentiation.