Résumé : Let $P$ be of the unitary group $U_A$ of a $C^*$-algebra $A$. The main result: in the von Neumann algebra context (i.e. if the isotropy sub-algebra is a von Neumann algebra), for each unit tangent vector $X$ at a point, there is a geodesic $\delta (t)$, wich is obtained by the action on $P$ of a $1$-parameter group in $U_A$. This geodesic is minimizing up to length $\pi /2.$