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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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    $\begingroup$ Could you say a bit more about the nature of the constraints? Are they algebraic equations in coordinates? $\endgroup$
    – S. Carnahan ♦
    Commented Aug 20, 2011 at 10:13
  • $\begingroup$ Yes, they are given as algebraic equations involving the $(x_1,...,x_n) \in \mathbb{R}^n$ and the functions of course. $\endgroup$
    – Mirco
    Commented Aug 20, 2011 at 10:30
  • $\begingroup$ ... Things like f(x,x)=f(-x,x). (This is just an example not a definitive required constraint) $\endgroup$
    – Mirco
    Commented Aug 20, 2011 at 10:33
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    $\begingroup$ I've never heard the term "n-velocity space" before. What is the precise definition? $\endgroup$
    – Deane Yang
    Commented Aug 20, 2011 at 13:14
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    $\begingroup$ I'm pretty sure the $n$-velocity space is isomorphic to the $n$-fold fiber product of $TM$ over $M$, or equivalently, the tensor product bundle of $TM$ with the trivial vector bundle of rank $n$. $\endgroup$
    – S. Carnahan ♦
    Commented Aug 20, 2011 at 16:01

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