The standard basic notions that are taught in the first course on Differential Geometry, such as: the notion of manifold, smooth maps, immersions and submersions, tangent vectors, Lie derivatives along vector fields, the flow of a vector field, the tangent space (and bundle), the definition of differential forms, de Rham operator and de Rham cohomology, integration and Stokes theorem. See e.g. the lecture notes of M. Crainic, or the book "Introduction to smooth manifolds" by John Lee.

Aim of the course
The aim of this course is to provide an introduction to vector bundles and principal bundle and, along the way,

illustrate their relevance to the study of various geometric structures on manifolds (Riemannian, foliations, etc).

The plan is to start with vector bundles and their geometry (connections, curvature, etc); when discussing metrics we will make a small digression into Riemannian geometry and the tubular neighborhood theorem. Then we move on to

principal bundles, their geometry, describe their relationship with vector bundles and their relevance to the theory of G-structures.

Along the way we will also improvise a crash course on Lie groups (only what we need).

Marius Crainic