Jet Bundle - Vector Fields

Vector Fields

A general vector field on the total space E, coordinated by, is

A vector field is called horizontal, meaning all the vertical coefficients vanish, if φα = 0.

A vector field is called vertical, meaning all the horizontal coefficients vanish, if ρi = 0.

For fixed (x, u), we identify

having coordinates (x, u, ρi, φα), with an element in the fiber T_xuE of TE over (x,u) in E, called a tangent vector in TE. A section

is called a vector field on E' with

and ψ in Γ(TE).

The jet bundle Jr(π) is coordinated by . For fixed (x,u,w), identify

having coordinates, with an element in the fiber of TJr(π) over (x, u, w) ∈ Jr(π), called a tangent vector in TJr(π). Here,

are real-valued functions on Jr(π). A section

is a vector field on Jr(π), and we say .