Existence and Uniqueness of Geodesics

Theorem
Let $M$ be a smooth manifold.

Let $TM$ be the tangent bundle of $M$.

Let $T_p M$ be the tangent space at $p \in M$.

Let $\nabla$ be a connection in $TM$.

Let $I \subseteq \R$ be an open real interval.

Let $\gamma : I \to M$ be a geodesic and $\gamma'$ its velocity.

Then:


 * $\forall p \in M : \forall w \in T_p M : \forall t_0 \in \R : \exists I \subseteq \R : t_0 \in I : \exists \gamma : \paren {\map \gamma {t_0} = p} \land \paren {\map {\gamma'} {t_0} = w}$

Furthermore, any two geodesics agree on their common domain.