Let $(M,g)$ be a riemannian manifold of dimension $n$ and $TN^{\perp}$ denote the normal bundle of a submanifold $N\subset M$. Now let's define $SN^{\perp}:=\{(x,v)\in TN^{\perp}: |v|=1\}, $ so this is a submanifold of $TM$ of dimension $n-1$. Let $\phi_t: TM\rightarrow TM$ denote the geodesic flow, and $\pi:TM\rightarrow M$.
Then in the book Geodesic flows the author introduces the function $A_N: SN^{\perp}\times \mathbb{R}\rightarrow \mathbb{R}$ by $A_N((x,v),t)\rightarrow |\det d_{(x,v)}(\pi\circ \phi_t)|_{T_{(x,v)}SN^{\perp}}|$.
Now I am not quite seeing why we are able to define this. To use the determinant we need that $ d_{(x,v)}(\pi\circ \phi_t)|_{T_{(x,v)}SN^{\perp}}$ maps to a subspace of dimension $n-1$ of $T_{x}M, $ but I can't see what this space actually is. How does one check that this function is well-defined?
Any help is appreciated. Thanks in advance.