In this lecture we begin the study of fibre bundles, focusing on the special case of vector bundles and principal bundles.

• The tangent bundle $\pi \colon TM \to M$ of a smooth manifold is an example of a vector bundle.
• The orbit map $\rho \colon M \to M/G$ is an example of a principal $G$-bundle.

Although it is not obvious from the definitions, the theories of vector bundles and principal bundles are essentially analogous, and it is largely a matter of taste whether one primarily works with vector bundles or principal bundles.

Roughly speaking: principal bundles are slightly more general, whereas vector bundles are slightly easier to understand. We will cover this in more detail next lecture.