Calabi-Yau structures on cotangent bundles

Starting with an orientable compact real-analytic Riemannian manifold $(L,g)$ with $\chi(L)=0$, we show that a small neighbourhood $\textrm{Op}(L)$ of the zero section in the cotangent bundle $T^{*}L$ carries a Calabi-Yau structure such that the zero section is an isometrically embedded special Lagrangian submanifold.