synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
For a smooth manifold and its cotangent bundle, there is a unique differential 1-form on itself,
with the property that under the isomorphism
between differential 1-forms and smooth sections of the cotangent bundle we have for every smooth section the identification
between the pullback of along and the 1-form corresponding to under .
This unique differential 1-form is called the Liouville form or Poincaré 1-form or canonical form or tautological form on the cotangent bundle.
The de Rham differential is a symplectic form. Hence every cotangent bundle is canonically a symplectic manifold.
On a coordinate chart of with canonical coordinate functions denoted , the cotangent bundle over the chart is with canonical coordinates . In these coordinates the canonical 1-form is (using Einstein summation convention)
and hence the symplectic form is
Last revised on August 28, 2024 at 16:38:04. See the history of this page for a list of all contributions to it.