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© 2003 by CRC Press LLC

meant to be symmetric with respect to the lower
indices µ ...µ for all 1 <h<k.
h
1
i
µ
Any section σ(x) = (x ,y (x)) induces a
J section of J B defined by
k
k µ i i i
j σ(x) = (x ,y (x), ∂ y (x),...,∂ y (x))
µ µ 1 ...µ k
jetbundle Let(B,M,π; F)beabundleand which is called the k-jet prolongation of σ.
N (π) be the set of local sections defined around
x Any bundle morphism ) : B → B (pro-
k
x ∈ M. The jet space at x is the space J B of
x jecting over a diffeomorphism φ : M → M )
k
equivalence classes, denoted by j σ,in N (π), k k k
x x induces a bundle morphism J ) : J B → J B
of sections having contact k at x (i.e., two local defined by
sections are equivalent if they have the same k-
order Taylor polynomial). The union of all such k k k −1
j )(j σ) = j f(x) () ◦ σ ◦ f )
x
jet spaces

k k
J B = J B which is called the k-jet prolongation of ).
x
x∈M
Analogously, any projectable vector field X
k
is a bundle over M (as well as over B and over all over B induces a projectable vector field j X
k
h
J B for h ≤ k), and it is called the k-jet prolon- over J B which is called the k-jet prolongation
µ
i
gation of B.If (x ; y ) are fibered coordinates of X.
µ
i
i
of B, then (x ,y ,y ,...,y i ) are fibered
µ µ 1 ...µ k
k
coordinates of J . The coordinates y i are junction point See branch point.
B µ 1 ...µ h

© 2003 by CRC Press LLC