I have been self-studying geodesic flows from Gabriel Paternain's book and I have a question about an exercise which I can't solve one of the implications :
I am asked to show that a vector $\eta\in T_{\theta}TM$ lies in $T_{\theta}SM$ if and only if $\langle K_{\theta}(\eta),v\rangle=0,$ where $\theta=(x,v)\in SM$.
The way the map $K_{\theta}$ is defined is as $K_{\theta}:T_{\theta}TM\rightarrow T_xM$, where $\theta=(x,v)\in TM$, and consider the curve $z(t)=(\alpha(t),Z(t))$ that give us $\eta$. Set $K_{\theta}(\eta)=\nabla_{\alpha}Z(0)$. One of the directions follows from differentiating, that is we have $\frac{d}{dt}\langle Z(t),Z(t)\rangle = 2 \langle \nabla_{\dot \alpha}Z, Z \rangle =0$, and using the fact that $Z(0)=v$.
For the other direction I can't quite figure it out . We know that $\langle K_{\theta}(\xi) , v \rangle =0$ and so this will tells us that $\frac{d}{dt}|_{t=0}\langle Z (t) , Z(t) \rangle =0$ and from the fact that $\theta \in SM$ we know that $\langle Z(0), Z(0) \rangle =1$ . However this is not enough to conclude that $\langle Z(t), Z(t) \rangle =1$ for any $t$ and so I don't know how to get that $\xi \in T_{\theta}SM$.
Thanks in advance.