The right-invariant metric on $SDiff\(M\)$ is not bi-invariant, so the geodesic from the identity element of SDiff(M) with an initial velocity field v should be different with the one-parameter subgroup with the same velocity field v. But along the geodesic connecting the identity and a diffeomorphism $\phi_{0t}^v\(x\)$, the Lagrangian and Eulerian velocty fields are related by the adjoint action $Ad_{\phi}$, it seems that the Eulerian velocity field along the geodesic is in fact right-invariant. If this is true, then the geodeisc is also a flow of a right-invariant vector field on SDiff(M) so that the geodesic and the one-parameter subgroup, i.e., the Riemannian exp and the Lie exp are the same. Is this true or there is a misunderstanding somewhere? thx.

Also a geodesic is a flow of the time-depedent Eulerian velocity field, which can just be regarded as part of a vector field on $SDiff\(M\)$(at least along the geodesic). why is the geodesic not a one-parameter subgroup(of course not necessary be the Lie exp)? There must be something wrong with my understanding. Please help me.