# A quantity associated with a Riemannian surface

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 $$M$$.