To expand Ryan Budney's comment, the geodesics of $(M,g)$ are the projection on the base $M$ of the integral curves for the vector field $S_g$ on $TM.$
$S_g$ is the unique vector field on $TM$ which is at the same time:
special (i.e. it represents a $2^\textrm{nd}$-order edo on $M$) M$, or equivalently its integral curves are the tangential lifting of their projection on the base) and horizontal (i.e. it is a section of the horizontal distribution on$TM$.) Because$S_g$is a spray (i.e.$\mathcal{L}_Z S_g=S_g,$where$Z$is the Euler vector filed), it is called the geodesic spray of$(M,g).$It can be realized even that$S_g$is the hamiltonian vector field of the kinetic energy$K_g:v\in TM\to\tfrac{1}{2}g(v,v)\in\mathbb{R}$with respect to the pull-back through$g^\flat$of the canonical symplectic form on$T^\ast M.$So$S_g$preserves the sphere bundles, and we get that if$M$is compact then$S_g$is complete. Quotienting $\textrm{pr}_1:\mathbb{R}\times TM\to\mathbb{R},$ under$F_\mathbb{R}\times TM\to TM,$Through the flow of$S_g,$we get its integral curves (which are the identification requested.tangential lifting of their projection on the base) are indentified with$TM\$.