We begin this lecture by introducing the Lie derivative associated to a vector field $X$. This is an operator $\mathcal{L}_X \colon \mathfrak{X}(M) \to \mathfrak{X}(M)$, defined by

$$\mathcal{L}_XY := \frac{\partial}{\partial t}\Big|_{t=0} ( \Phi_{-t})_\ast Y,$$

where $\Phi_t$ is the flow of $X$. In fact, this operator is nothing new, for we prove that the Lie derivative agrees with the Lie bracket:

$$\mathcal{L}_X Y = [X,Y].$$

We then move onto our next major topic. Over the following four lectures we will cover the basic theory of Lie groups. Lie groups are important in many areas of mathematics (not just geometry!)—including representation theory, harmonic analysis, differential equations and more. Lie groups also crop up naturally in physics—both classically (eg. Noether's theorem that every smooth symmetry of a physical system has a corresponding conservation law), and in high-energy particle physics, via gauge theory. We'll touch upon gauge theory in Differential Geometry II when we study connections on principal bundles.

Today we got as far as stating the famous Closed Subgroup Theorem, which says that a closed subgroup of a Lie group is automatically a Lie subgroup.