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.