For submersion and submetry,why can we lift a geodesic "horizontally" to a geodesic?

Question

A map $\sigma:X\to Y$ between locally compact complete inner metric spaces is called a submetry if $\sigma(B_r(p))=B_r(\sigma(p))$ for all $r>0$ and $p\in X$. Why is that a geodesic in $Y$ can be lifted "horizontally" to a geodesic in $X$. And the case for submersion? The problem lies when we lift a point we don't know where we lift in the fiber.