Editing Jet (mathematics) (section)

== Jets of functions between two manifolds ==

If ''M'' and ''N'' are two [[differentiable manifold|smooth manifolds]], how do we define the jet of a function <math>f:M\rightarrow N</math>? We could perhaps attempt to define such a jet by using [[manifold|local coordinates]] on ''M'' and ''N''. The disadvantage of this is that jets cannot thus be defined in an invariant fashion. Jets do not transform as [[tensors]]. Instead, jets of functions between two manifolds belong to a [[jet bundle]].

===Jets of functions from the real line to a manifold===
Suppose that ''M'' is a smooth manifold containing a point ''p''. We shall define the jets of [[curve]]s through ''p'', by which we henceforth mean smooth functions <math>f:{\mathbb R}\rightarrow M</math> such that ''f''(0)&nbsp;=&nbsp;''p''. Define an equivalence relation <math>E_p^k</math> as follows. Let ''f'' and ''g'' be a pair of curves through ''p''. We will then say that ''f'' and ''g'' are equivalent to order ''k'' at ''p'' if there is some [[neighborhood (mathematics)|neighborhood]] ''U'' of ''p'', such that, for every smooth function <math>\varphi : U \rightarrow {\mathbb R}</math>, <math>J^k_0 (\varphi\circ f)=J^k_0 (\varphi\circ g)</math>. Note that these jets are well-defined since the composite functions <math>\varphi\circ f</math> and <math>\varphi\circ g</math> are just mappings from the real line to itself. This equivalence relation is sometimes called that of ''k''-th-order [[contact (mathematics)|contact]] between curves at ''p''.

We now define the '''''k''-jet''' of a curve ''f'' through ''p'' to be the equivalence class of ''f'' under <math>E^k_p</math>, denoted <math>J^k\! f\,</math> or <math>J^k_0f</math>. The '''''k''-th-order jet space''' <math>J^k_0({\mathbb R},M)_p</math> is then the set of ''k''-jets at ''p''.
 
As ''p'' varies over ''M'', <math>J^k_0({\mathbb R},M)_p</math> forms a [[fibre bundle]] over ''M'': the ''k''-th-order [[tangent bundle]], often denoted in the literature by ''T''<sup>''k''</sup>''M'' (although this notation occasionally can lead to confusion). In the case ''k''=1, then the first-order tangent bundle is the usual tangent bundle: ''T''<sup>1</sup>''M''&nbsp;=&nbsp;''TM''.

To prove that ''T''<sup>''k''</sup>''M'' is in fact a fibre bundle, it is instructive to examine the properties of <math>J^k_0({\mathbb R},M)_p</math> in local coordinates. Let (''x''<sup>''i''</sup>)= (''x''<sup>1</sup>,...,''x''<sup>''n''</sup>) be a local coordinate system for ''M'' in a neighborhood ''U'' of ''p''. [[abuse of notation|Abusing notation]] slightly, we may regard (''x''<sup>''i''</sup>) as a local [[diffeomorphism]] <math>(x^i):M\rightarrow\R^n</math>.

''Claim.'' Two curves ''f'' and ''g'' through ''p'' are equivalent modulo <math>E_p^k</math> if and only if <math>J^k_0\left((x^i)\circ f\right)=J^k_0\left((x^i)\circ g\right)</math>.

:Indeed, the ''only if'' part is clear, since each of the ''n'' functions ''x''<sup>1</sup>,...,''x''<sup>''n''</sup> is a smooth function from ''M'' to <math>{\mathbb R}</math>. So by the definition of the equivalence relation <math>E_p^k</math>, two equivalent curves must have <math>J^k_0(x^i\circ f)=J^k_0(x^i\circ g)</math>.

:Conversely, suppose that <math>\varphi</math>; is a smooth real-valued function on ''M'' in a neighborhood of ''p''. Since every smooth function has a local coordinate expression, we may express <math>\varphi</math>; as a function in the coordinates. Specifically, if ''q'' is a point of ''M'' near ''p'', then

::<math>\varphi(q)=\psi(x^1(q),\dots,x^n(q))</math>

:for some smooth real-valued function &psi; of ''n'' real variables. Hence, for two curves ''f'' and ''g'' through ''p'', we have

::<math>\varphi\circ f=\psi(x^1\circ f,\dots,x^n\circ f)</math>
::<math>\varphi\circ g=\psi(x^1\circ g,\dots,x^n\circ g)</math>

:The chain rule now establishes the ''if'' part of the claim. For instance, if ''f'' and ''g'' are functions of the real variable ''t'' , then

::<math>\left. \frac{d}{dt} \left( \varphi\circ f \right) (t) \right|_{t=0}= \sum_{i=1}^n\left.\frac{d}{dt}(x^i\circ f)(t)\right|_{t=0}\ (D_i\psi)\circ f(0)</math>

:which is equal to the same expression when evaluated against ''g'' instead of ''f'', recalling that ''f''(0)=''g''(0)=p and ''f'' and ''g'' are in ''k''-th-order contact in the coordinate system (''x''<sup>''i''</sup>).

Hence the ostensible fibre bundle ''T''<sup>''k''</sup>''M'' admits a local trivialization in each coordinate neighborhood. At this point, in order to prove that this ostensible fibre bundle is in fact a fibre bundle, it suffices to establish that it has non-singular transition functions under a change of coordinates. Let <math>(y^i):M\rightarrow{\mathbb R}^n</math> be a different coordinate system and let <math>\rho=(x^i)\circ (y^i)^{-1}:{\mathbb R}^n\rightarrow {\mathbb R}^n</math> be the associated [[change of coordinates]] diffeomorphism of Euclidean space to itself. By means of an [[affine transformation]] of <math>{\mathbb R}^n</math>, we may assume [[without loss of generality]] that ρ(0)=0. With this assumption, it suffices to prove that <math>J^k_0\rho:J^k_0({\mathbb R}^n,{\mathbb R}^n)\rightarrow J^k_0({\mathbb R}^n,{\mathbb R}^n)</math> is an invertible transformation under jet composition. (See also [[jet group]]s.) But since ρ is a diffeomorphism, <math>\rho^{-1}</math> is a smooth mapping as well. Hence,

:<math>I=J^k_0I=J^k_0(\rho\circ\rho^{-1})=J^k_0(\rho)\circ J^k_0(\rho^{-1})</math>

which proves that <math>J^k_0\rho</math> is non-singular. Furthermore, it is smooth, although we do not prove that fact here.

Intuitively, this means that we can express the jet of a curve through ''p'' in terms of its Taylor series in local coordinates on ''M''.

''Examples in local coordinates:''

* As indicated previously, the 1-jet of a curve through ''p'' is a tangent vector. A tangent vector at ''p'' is a first-order [[differential operator]] acting on smooth real-valued functions at ''p''. In local coordinates, every tangent vector has the form

::<math>v=\sum_iv^i\frac{\partial}{\partial x^i}</math>

:Given such a tangent vector ''v'', let ''f'' be the curve given in the ''x''<sup>''i''</sup> coordinate system by <math>x^i\circ f(t)=tv^i</math>. If ''&phi;'' is a smooth function in a neighborhood of ''p'' with ''&phi;''(''p'')&nbsp;=&nbsp;0, then

::<math>\varphi\circ f:{\mathbb R}\rightarrow {\mathbb R}</math>

:is a smooth real-valued function of one variable whose 1-jet is given by

::<math>J^1_0(\varphi\circ f)(t)=\sum_itv^i \frac{\partial \varphi}{\partial x^i}(p).</math>

:which proves that one can naturally identify tangent vectors at a point with the 1-jets of curves through that point.

* The space of 2-jets of curves through a point.
: In a local coordinate system ''x<sup>i</sup>'' centered at a point ''p'', we can express the second-order Taylor polynomial of a curve ''f''(''t'') through ''p'' by

::<math>J_0^2(x^i(f))(t)=t\frac{dx^i(f)}{dt}(0)+\frac{t^2}{2}\frac{d^2x^i(f)}{dt^2}(0).</math>

:So in the ''x'' coordinate system, the 2-jet of a curve through ''p'' is identified with a list of real numbers <math>(\dot{x}^i,\ddot{x}^i)</math>. As with the tangent vectors (1-jets of curves) at a point, 2-jets of curves obey a transformation law upon application of the coordinate transition functions.

:Let (''y''<sup>''i''</sup>) be another coordinate system. By the chain rule,

::<math>
\begin{align}
\frac{d}{dt}y^i(f(t)) & = \sum_j\frac{\partial y^i}{\partial x^j}(f(t))\frac{d}{dt}x^j(f(t)) \\[5pt]
\frac{d^2}{dt^2}y^i(f(t)) & = \sum_{j,k}\frac{\partial^2 y^i}{\partial x^j \, \partial x^k}(f(t))\frac{d}{dt}x^j(f(t)) \frac{d}{dt}x^k(f(t))+\sum_j\frac{\partial y^i}{\partial x^j}(f(t))\frac{d^2}{dt^2}x^j(f(t))
\end{align}
</math>

:Hence, the transformation law is given by evaluating these two expressions at ''t''&nbsp;=&nbsp;0.

::<math>
\begin{align}
& \dot{y}^i=\sum_j\frac{\partial y^i}{\partial x^j}(0)\dot{x}^j \\[5pt]
& \ddot{y}^i=\sum_{j,k}\frac{\partial^2 y^i}{\partial x^j \, \partial x^k}(0)\dot{x}^j\dot{x}^k+\sum_j\frac{\partial y^i}{\partial x^j}(0)\ddot{x}^j.
\end{align}
</math>

:Note that the transformation law for 2-jets is second-order in the coordinate transition functions.

===Jets of functions from a manifold to a manifold===
We are now prepared to define the jet of a function from a manifold to a manifold.

Suppose that ''M'' and ''N'' are two smooth manifolds. Let ''p'' be a point of ''M''. Consider the space <math>C^\infty_p(M,N)</math> consisting of smooth maps <math>f:M\rightarrow N</math> defined in some neighborhood of ''p''. We define an equivalence relation <math>E^k_p</math> on <math>C^\infty_p(M,N)</math> as follows. Two maps ''f'' and ''g'' are said to be ''equivalent'' if, for every curve γ through ''p'' (recall that by our conventions this is a mapping <math>\gamma:{\mathbb R}\rightarrow M</math> such that <math>\gamma(0)=p</math>), we have <math>J^k_0(f\circ \gamma)=J^k_0(g\circ \gamma)</math> on some neighborhood of ''0''.

The jet space <math>J^k_p(M,N)</math> is then defined to be the set of equivalence classes of <math>C^\infty_p(M,N)</math> modulo the equivalence relation <math>E^k_p</math>. Note that because the target space ''N'' need not possess any algebraic structure, <math>J^k_p(M,N)</math> also need not have such a structure. This is, in fact, a sharp contrast with the case of Euclidean spaces.

If <math>f:M\rightarrow N</math> is a smooth function defined near ''p'', then we define the ''k''-jet of ''f'' at ''p'', <math>J^k_pf</math>, to be the equivalence class of ''f'' modulo <math>E^k_p</math>.

===Multijets===
[[John Mather (mathematician)|John Mather]] introduced the notion of ''multijet''. Loosely speaking, a multijet is a finite list of jets over different base-points. Mather proved the multijet [[transversality theorem]], which he used in his study of [[stable mapping]]s.