In this lecture we give an axiomatic description of a parallel transport system on a vector bundle.

Next lecture we will prove that a connection on a vector bundle determines (and is uniquely determined by) a parallel transport system, and thus the two concepts can be used interchangeably.

In essence, a parallel transport system $\mathbb{P}$ on a vector bundle $E \to M$ allows us to associate to any curve $\gamma \colon [a,b] \to M$ a linear isomorphism from $E_{ \gamma(a)}$ to $E_{ \gamma(b)}$. Whilst this isomorphism depends on the path $\gamma$, it gives us a way to identify the vector space $E_{ \gamma(a)}$ with the vector space $E_{ \gamma(b)}$. As explained at the start of the last lecture, this will allow us to make sense of “differentiating a section along a vector field”, which ultimately is what connections are designed to achieve.