The k-th order jet groupGnk consists of jets of smooth diffeomorphisms φ: Rn → Rn such that φ(0)=0.[1]
The following is a more precise definition of the jet group.
Let k ≥ 2. The differential of a function f:Rk → R can be interpreted as a section of the cotangent bundle of RK given by df:Rk → T*Rk. Similarly, derivatives of order up to m are sections of the jet bundleJm(Rk) = Rk × W, where
Here R* is the dual vector space to R, and Si denotes the i-th symmetric power. A smooth function f:Rk → R has a prolongation jmf: Rk → Jm(Rk) defined at each point p ∈ Rk by placing the i-th partials of f at p in the Si((R*)k) component of W.
Consider a point . There is a unique polynomial fp in k variables and of order m such that p is in the image of jmfp. That is, . The differential data x′ may be transferred to lie over another point y ∈ Rn as jmfp(y) , the partials of fp over y.
Provide Jm(Rn) with a group structure by taking
With this group structure, Jm(Rn) is a Carnot group of class m + 1.
^Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), (PDF), Springer-Verlag, pp. 128–131, archived from the original (PDF) on 2017-03-30, retrieved 2014-05-02.
Referencesedit
Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), (PDF), Springer-Verlag, archived from the original (PDF) on 2017-03-30, retrieved 2014-05-02
Krupka, Demeter; Janyška, Josef (1990), Lectures on differential invariants, Univerzita J. E. Purkyně V Brně, ISBN80-210-0165-8
Saunders, D.J. (1989), The geometry of jet bundles, Cambridge University Press, ISBN0-521-36948-7
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group, mathematics, group, generalization, general, linear, group, which, applies, taylor, polynomials, instead, vectors, point, group, group, jets, that, describes, taylor, polynomial, transforms, under, changes, coordinate, systems, equivalently, diffeomorph. In mathematics a jet group is a generalization of the general linear group which applies to Taylor polynomials instead of vectors at a point A jet group is a group of jets that describes how a Taylor polynomial transforms under changes of coordinate systems or equivalently diffeomorphisms Overview editThe k th order jet group Gnk consists of jets of smooth diffeomorphisms f Rn Rn such that f 0 0 1 The following is a more precise definition of the jet group Let k 2 The differential of a function f Rk R can be interpreted as a section of the cotangent bundle of RK given by df Rk T Rk Similarly derivatives of order up to m are sections of the jet bundle Jm Rk Rk W where W R R k S 2 R k S m R k displaystyle W mathbf R times mathbf R k times S 2 mathbf R k times cdots times S m mathbf R k nbsp Here R is the dual vector space to R and Si denotes the i th symmetric power A smooth function f Rk R has a prolongation jmf Rk Jm Rk defined at each point p Rk by placing the i th partials of f at p in the Si R k component of W Consider a point p x x J m R n displaystyle p x x in J m mathbf R n nbsp There is a unique polynomial fp in k variables and of order m such that p is in the image of jmfp That is j k f p x x displaystyle j k f p x x nbsp The differential data x may be transferred to lie over another point y Rn as jmfp y the partials of fp over y Provide Jm Rn with a group structure by taking x x y y x y j m f p y y displaystyle x x y y x y j m f p y y nbsp With this group structure Jm Rn is a Carnot group of class m 1 Because of the properties of jets under function composition Gnk is a Lie group The jet group is a semidirect product of the general linear group and a connected simply connected nilpotent Lie group It is also in fact an algebraic group since the composition involves only polynomial operations Notes edit Kolar Ivan Michor Peter Slovak Jan 1993 Natural operations in differential geometry PDF Springer Verlag pp 128 131 archived from the original PDF on 2017 03 30 retrieved 2014 05 02 References editKolar Ivan Michor Peter Slovak Jan 1993 Natural operations in differential geometry PDF Springer Verlag archived from the original PDF on 2017 03 30 retrieved 2014 05 02 Krupka Demeter Janyska Josef 1990 Lectures on differential invariants Univerzita J E Purkyne V Brne ISBN 80 210 0165 8 Saunders D J 1989 The geometry of jet bundles Cambridge University Press ISBN 0 521 36948 7 nbsp This algebra related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Jet group amp oldid 1005351905, wikipedia, wiki, book, books, library,