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Is the following fact well known?

Let $X$ be a manifold and $V$ be a vector space. Let $E_1$ be a sub-bundle of the constant bundle $X \times V$. Let $E_2$ be its orthogonal compliment in $X \times V^*$. Then the co-normal bundle of the total space of $E_1$ inside $X \times V$ coincides with the co-normal bundle of the total space of $E_2$ inside $X \times V^* $, under the identification $T^* (X \times V)=T^* X \times V \times V^* =T^* (X \times V^* )$.

I know the proof, but I prefer not to write it in a paper if it is a well known fact.

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I suppose you mean $E_2$ is the annihilator of $E_1$, since it is a subbundle of $X \times V^*$, and since you don't have a metric on $X \times V$ anyway. It's not a long proof, so what does it hurt to write it in the paper if it helps the reader? The worst than can happen is that the referee will ask you to remove it. I wouldn't do that if I was the referee. –  Spiro Karigiannis Feb 5 '12 at 1:46

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