Let M be a Riemannian manifold $p\in M$, $X,Y\in T_pM$. I am given the variation $v(s,t)=exp_p(t(X+s\ Y))$ of $\gamma(t)=exp_p(t\ X)$.
Our Professor gives us in the discussion the identity $$||\frac{\partial v(s,t)}{\partial t}||_{g_{v(s,t)}}=||\frac{\partial v(s,t)}{\partial t}||_{g_{v(0,t)}}=||X+sY||_p$$
I see that $||\frac{\partial v(s,t)}{\partial t}||_{g_{v(s,t)}}=||X+sY||_p$, because $v$ is in $t$ direction a geodesic and the geodesic has to be unit speed (Edit: constant speed, not necessarily unit speed). So the velocity at $t$ equals the starting condition.
The step in the middle seems strange. Could somebody verify if the middle step is correct and why? Shouldn't $\frac{\partial v(s,t)}{\partial t}$ live at $T_{v(s,t)}M$.