Cotangent Bundles

In this section, we denote by P a manifold, and denote the cotangent bundleCotangent bundle on P by $$T^{*}P$$ . The fiber $$T^{*}_xP$$ of $$T^{*}P$$ at any point $$x\in P$$ is the dual space of the vector space $$T_xP$$ , and the elements in $$T^{*}_xP$$ are the cotangent vectors at the point x. We use $$\pi $$ and $$\pi _{*}$$ to denote the projections of TP and $$T^{*}P$$ on P respectively. We use $$T(T^{*}P)$$ to denote the tangent bundle of the cotangent bundle $$T^{*}P$$ and use $$\pi _0$$ to denote the projection of $$T(T^{*}P)$$ on the base space $$T^{*}P$$ .