Cotangent Bundles

Abstract

In this section, we denote by P a manifold, and denote the cotangent 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 $T^{*}P$.