**A. Pereira do Vale & M. R. Pinto (ed.),
**

Proceedings of the 1st International Meeting on Geometry and Topology.

Proceedings of the conference held in Braga, September 11-13, 1997.

Universidade do Minho, Portugal

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Explicit Formulas for the 3-Jet Lift of a Matrix Group. Apllications to Conformal Geometry

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E. Aguirre-Dab and I. Sanchez-Rodriguez

Departamento de Geometria y Topologia, Facultad de Matematicas, Universidad Complutense, 28040 Madrid, Spain Eduardo_Aguirre@Mat.UCM.Es
Departamento de Geometria y Topologia, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain ignacios@goliat.ugr.es

**Abstract:** The 3-jet lift $G^3$ of a matrix group $G$ is isomorphic, via a map that we give explicitly, to a semidirect product of $G$ itself and a nilpotent group builded up from the first two prolongations of its Lie algebra. Using this isomorphism, we write down the formulas for the most natural representations of $G^3$, as well as for one additional representation of the 2-jet lift $G^2$ appearing when $G$ is of finite type 2. We apply these results to the case of the (linear) conformal group and we point out the geometric implications of these representations.

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