Description:
This course will cover the fundamentals of differential geometry
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.