Assume that $E$ is a Riemannian vector bundle, then its structure group is reduced to $O(n)$. Then the structure group of $E \oplus E$ is reduced to $D(O(n) \oplus O(n)) \subset Sp(2n)$ where $$D(O(n) \oplus O(n))=\{ A\oplus A\mid A \in O(n)\}$$
So there is a natural symplectic structure $\omega $ on each fiber of $E \oplus E$.(We use a maximal atlas of trivialization charts for $E \oplus E$ whose structure group is contained in $D(O(n) \oplus O(n))$.
let $T:E \to E \oplus E$ be an arbitrary morphism. We pull back $\omega $ to $T^*(\omega)$. So $T^*(\omega)$ is a $2$- form on each fiber of $E$.
The above construction can be applied to $E=TM$ where $M$ is a compact Riemannian $2$ dimensional manifold.
Associated with a compact Riemannian surface $(M,g)$, we define a a quantity $q(M,g)$ as follows: $$q(M,g) =\sup_T |\int_M T^*(\omega)|$$ where $T$ varies on all isometric embedding of $TM$ into $TM \oplus TM$ where the latter is equipped with the product metric.
Is this quantity finite? Is it independent of choosing a maximal trivialization atlas for $TM \oplus TM$ whose transition maps are contained in $D(O(n) \oplus O(n))$?
If the answer of the above question is "yes", what would be an interpretation of this quantity $q(M,g)$?
Note that the same construction can be repeated for an arbitrary compact $2k$ dimensional manifold by integration of ${(T^*(\omega))}^k$ on $M$.
