In this lecture we define the exponential map of $\exp$ associated to a spray $\mathbb{S}$ on a manifold $M$ via

$$\exp_p(\xi) := \pi ( \Phi_1(p,\xi) ), \qquad p \in M, \ \xi \in \mathscr{E}_p,$$

where $\Phi_t$ is the maximal flow of $\mathbb{S}$ and $\mathscr{E}_p \subset T M$ is a starshaped open set on which $\Phi_1$ is defined. We show that $\exp$ is a diffeomorphism on a neighbourhood of the zero section.

We then turn our attention to the converse of the main theorem from the previous lecture, which states that if $\mathbb{S}$ is a spray on $M$ then there exists a connection $\nabla$ for which $\mathbb{S}$ becomes the geodesic spray of $\nabla$.

This connection $\nabla$ is not unique — whilst for each connection there is a unique geodesic spray, in general many different connections can have the same geodesics (and hence the same geodesic spray). We will rectify this annoyance next lecture when we introduce the notion of a torsion-free connection.