Cotangent Bundles

  • Jean-Louis KoszulEmail author
  • Yi Ming Zou


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 \(T^{*}P\).

Copyright information

© Springer Nature Singapore Pte Ltd. and Science Press 2019

Authors and Affiliations

  1. 1.Institut FourierUniversité Grenoble AlpesGières, GrenobleFrance
  2. 2.Department of Mathematical SciencesUniversity of Wisconsin-MilwaukeeMilwaukeeUSA

Personalised recommendations