# 43. The Ambrose-Palais-Singer Spray Theorem

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

$$ \exp^{ \mathbb{S}}_x(v) := \pi ( \Theta^{ \mathbb{S}}_1(v) ), \qquad x \in M, \ v \in \mathcal{S}_x,$$

where $ \Theta^{\mathbb{S}}$ is the maximal flow of $ \mathbb{S}$ and $ \mathcal{S}_x \subset T M$ is a starshaped open set on which $ \Theta_1^{ \mathbb{S}}$ is defined. We show that $ \exp^{ \mathbb{S}}$ is a diffeomorphism on a neighbourhood of the zero section.

We then turn our attention to the *Ambrose-Palais-Singer Spray Theorem*, 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.

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