Maybe this is a silly question (or not even a question), but I was wondering whether the cotangent bundle of a submanifold is somehow **canonically** related to the cotangent bundle of the ambient space.
To be more precise:<br>
Let $N$ be a manifold and $\iota:M \hookrightarrow  N$ be an embedded (immersed) submanifold. Is the cotangent bundle $T^\ast M$ somehow canonical related to the cotangent bundle $T^\ast N$. Canonical means, without choosing a metric on $N$. The choice of a metric gives an isomorphism of $TN$ and $T^\ast N$ and therefore a "relation", since the tangent bundle of the submanifold $M$ can be viewed in a natural way as a subspace of the tangent bundle of the ambient space $N$ ($\iota$ induces an injective linear map at each point $\iota_\ast : T_pM \rightarrow T_pN$). I think this is not true for the cotangent space (without a metric)<br>
Moreover, the cotangent bundle $T^\ast N$ of a manifold $N$ is a kind of "prototype" of a symplectic manifold. The symplectic structure on $T^\ast N$ is given by $\omega_{T^\ast N} = -d\lambda$, where $\lambda$ is the Liouville form on the cotangent bundle. (tautological one-form, canonical one-form, symplectic potential or however you want). The cotangent bundle of the submanifold $T^\ast M$ inherits in the same way a canonical symplectic structure. So, is there a relation between $T^\ast N$ and $T^\ast M$ respecting the canonical symplectic structures. (I think the isomorphism given by a metric is respecting (relating) these structures, or am I wrong?) As I said, this question is perhaps strange, but the **canonical** existence of the symplectic structure on the cotangent bundle is "quite strong". For example:<br>
A given diffeomorphism $f:X \rightarrow Y$ induces a canonical symplectomorphism $T^\ast f : T^\ast Y \rightarrow T^\ast X$ (this can be proved by the special "pullback cancellation" property of the Liouville form).
So in the case of a diffeomorphism the symplectic structures are "the same". 
Ok, a diffeomorphism has more structure than an embedding, but perhaps there is a similar relation between $T^\ast M$ and $T^\ast N$?