We will begin by reviewing the inverse and implicit function theorems, introducing differentiable manifolds, defining and deriving the basic properties of differential forms and vector fields, and proving the flow box and Frobenius theorems. Along the way, we will discuss Sard's theorem and its applications. We will also construct many examples of differentiable manifolds as they arise in various contexts.
We will then define vector bundles, metrics, and connections and their curvatures. We will then study geodesics and curvature.
We will examine some relations between curvature and topology.
Applications to advanced topics will be considered if time permits.