Domain and exponential map of a connection

Every tangent vector $v\in TM$ has the unique base point $p=\pi (v)$, and [F1] uniquely determines both $I_{p,v}$ and ${\gamma }_{p,v}$. Thus membership in $\mathcal{E}$ and the value ${\gamma }_{p,v}(1)$ are well-defined. The definition only evaluates curves whose maximal interval actually contains $1$ and therefore does not presume geodesic completeness.

The zero vector $0_p$ gives the constant geodesic on all of $\mathbb{R}$, so $0_p\in {\mathcal{E}}_p$ and ${\exp }_p(0_p)=p$. In dimension zero all tangent vectors are zero; if $M$ is empty, then $TM$ and $\mathcal{E}$ are empty and the displayed map is the unique empty function. Because $I_{p,v}$ is open, $1\in I_{p,v}$ is an interior-time condition rather than an included-endpoint convention; $\mathcal{E}$ may still be a proper subset of $TM$. The only choice principle used is the stated ${\mathrm{A}\mathrm{C}}_{\omega }$ inherited through [F1].