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Let $(E,M,p)$ be a n dimensinal smooth vector bundle where $M$ is a k dimensional manifold. We assign to $M$, two different vector bundles $F_{1}$ and $F_{2}$ over $M$ as follows:

1)$TE$ is a vector bundle over $E$ and $E$ contains a copy of $M$ as the zero section: We define $F_{1}$= the restriction of $TE$ to $M$ as the zero section of $E$. So $F_{1}$ is a n+k dimensional vector bundle over $M$

2)$(TE, TM, Dp)$ has a natural structure of a vector bundle: we denote by $F_{2}$, the restriction of this bundle structure to $M\subset TM$, as the zero section of $TM$. So $F_{2}$ is a 2n dimensional vector bundle over $M$.

Question:

Is $F_{2}$ stably equivalent to $F_{1}$?

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It seems to me that $F_1=E\oplus TM$ and $F_2=E\oplus E$ (Whitney sums). These decompositions are not quite canonical: there are short exact sequences, but in this category they always split. Anyway, it follows that $F_1$ and $F_2$ are stably equivalent iff so are $E$ and $TM$.

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