Here is my second question on understanding jets better:
For a smooth manifold $M$ the set of jets $J^1_0(\mathbb{R},M)$ is the same as the Tangent bundle $TM$. This implies that any equivalence class $j_0^1f$ of curves $f:\mathbb{R} \rightarrow M$ is a tangent vector at $f(0)$.
Now suppose $N \subset M$ is a submanifold of $M$. Then we can look on it from two different angles:
First $N$ is a manifold in its own right. In that case the jet class $j_0^1f \in J^1_0(\mathbb{R},N)$ contains smooth maps $f: \mathbb{R} \rightarrow N$ only.
But if we look on $N$ as a submanifold of $M$, then a jet class $j_0^1g \in J^1_0(\mathbb{R},M)$, that is tangent to $N$, defining the same tangent vector as $j_0^1f$, is not 'the same' as $j_0^1f$ as a set of curves.
Although $j_0^1g$ is tangential to $N$, it contains curves that intersect $N$ only in $g(0)$ just having the same first order derivative in $g(0)$.
So seen this way, although $j_0^1g$ and $j^0_1f$ define the same tangent vector in $f(0)$ they are not the same as sets of curves.
Is this right?
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Now to show that a jet $j_0^1g \in J^1_0(\mathbb{R},M)$ is actually tangent to a submanifold $N$, is it sufficient to show that there is a representative $h \in j^1_0g$ and an open neighbourhood $U$ of zero in $\mathbb{R}$ such that $h$ is locally a curve in $N$, that is $h : U \rightarrow N$ ?