Let M be a hyperbolic 3-manifold, let $g_{t}$ be a smooth family of hyperbolic metrics on M. It is known that in this case we can find a family of devlopping maps $dev_{g_{t}}:\tilde{M} \to \mathbb{H}^3$ which is smooth in t, namely $\forall x \in \tilde{M}$, $t \mapsto dev_{g_{t}}(x)$ is smooth.
An e example of such family is the following: fix $x_{0} \in \mathbb{H}^3$, $p_{0} \in \tilde{M}$, and an isometry $I:T_{p_{0}}\tilde{M} \to T_{x_{0}}\mathbb{H}^{3}$ and assume that $ \forall t$ $dev_{g_{t}}(p_{0}) = x_{0}$ and $d_{p_{0}}dev_{g_{t}}= I$.
My question is: -why the last example is correct ?
A known fact is that any isometry between two connected complete Riemannian manifolds can uniquely be determined by its image at one point and its differential at that point (I think probably the exemple follows from this fact but I don't see how)
