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Lets be in the category $\mathbf{M}$ of smooth finite dimensional manifolds with smooth maps:

Suppose we have the set of all smooth maps $Hom_\mathbf{M}(R^n,M)$ from $R^n$ to a smooth manifold $M$. Taking only the first Taylor-Approximation of any such map, gives us the n-velocity Space $T_nM$, that is a fiber bundle over $M$ and a generalization of $TM$.

Now suppose we just have a subset of $Hom(R^n,M)$ given by some constraints on the maps lets say $$Hom_\mathbf{M}( R^n,M | constraint_1(f) ... constraint_n(f) )$$.

Does this always gives us a subbundle of $T_nM$?

If not, what should be required to be a subbundle?

Or say it the other way around: What must we proof, to show that the appropriate Jet set is a subbundle of $T_nM$?

(For my purposes its enough to suppose, that the constraints are of zero-order i.e. don't involve any derivations)

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# Jet spaces for maps with constraints

Lets be in the category $\mathbf{M}$ of smooth finite dimensional manifolds with smooth maps:

Suppose we have the set of all smooth maps $Hom_\mathbf{M}(R^n,M)$ from $R^n$ to a smooth manifold $M$. Taking only the first Taylor-Approximation of any such map, gives us the n-velocity Space $T_nM$, that is a fiber bundle over $M$ and a generalization of $TM$.

Now suppose we just have a subset of $Hom(R^n,M)$ given by some constraints on the maps lets say $$Hom_\mathbf{M}( R^n,M | constraint_1(f) ... constraint_n(f) )$$.

Does this always gives us a subbundle of $T_nM$?

If not, what should be required to be a subbundle?

Or say it the other way around: What must we proof, to show that the appropriate Jet set is a subbundle of $T_nM$?