This comes from page 125 in the chapter on isometric immersions.
Setup:
$f:M \rightarrow \bar{M}$ is an immersion. For each $p \in M$ there exists a neibhborhood $U \subset M$ of $p$ such that $f(U) \subset \bar{M}$ is a submanifold of $\bar{M}$.
He says to simplify the notation we will identify $U$ with $f(U)$ and each $v \in T_q M$, $q \in U$ with $df_q v \in T_{f(q)}\bar{M}$.
Here is where I get confused:
He says, for each $p \in M$, the inner product on $T_p \bar{M}$ splits $T_p\bar{M}$ into the direct sum $T_p\bar{M}=T_pM \oplus (T_pM)^{\perp}$.
Further, if $v \in T_p \bar{M}$, $p \in M$, we can write $v=v^{T}+v^{N}$ where $v^T \in T_p M$ and $v^N \in (T_p M)^\perp$. He says such a splitting is clearly differentiable, in the sense that the mappings $(p,v) \mapsto (p,v^T)$ and $(p,v)\mapsto (p,v^N)$ of $T\bar{M}$ into $T\bar{M}$ are clearly differentiable.
My issue with this:
I think every instance of $T_p \bar{M}$ should be replaced by $T_{f(p)}\bar{M}$,(i.e. we identify $T_p M$ with $df_p(T_p M) \subset T_f(p)\bar{M})$, unless
if $p \in U$ then we could make the identifications that he stated which would make more sense, however, he says this for $p \in M$, not just $p \in U$, and
He made the "mistake" of using $p$ several times which makes me believe that he meant to.
So what is going on here? Is he just being incredibly sloppy or am I missing something here?