Let's say we denote by $T^{(n,m)}M$ the vector-bundle of rank $(n,m)$ tensors on a manifold $M$ and by $\Lambda^pM$ the vector-bundle of $p$-forms on $M$. Is there a relationship (perhaps a diffeomorphism to some sort of direct sum expansion) between the various $\Lambda^p M$ (or the various $T^{(n,m)}M$) and iterated $TM$'s or $T^*M$'s? Intuitively, for example, the tensors of rank $2$ are like 2nd order terms in a Taylor expansion -- but so is $TTM$. Using physics notation, where $M$ is parametrized locally by coordinates $q$, $TM$ looks like $(q,\dot{q})$ and $TTM$ look like $(q,\dot{q},\dot{q}',\ddot{q})$ where I use the ' to denote a 2nd copy of the $\dot{q}$. If $M$ is $r$ dimensional then $TM$ has dimension $2r$, $TTM$ has dimension $4r$, etc, and $\Lambda^p M$ has dimension $({r\atop p})+r$. Is there some sort of useful morphism between $\Lambda^pM$ and a direct sum of $T^*M$, $T^*T^*M$, etc? Ditto for the $T^{(n,m)}M$? It feels like there should be some relationship between tensor bundles (or form-bundles) and iterated tangent or cotangent bundles, but I can't seem to find one (or determine an obvious ways to construct a series that even would have the correct dimension). On a similar note, are there any such relationships amongst the iterated $TM$'s and $T^*M$'s themselves? Thanks in advance for any insights.