# Reductive G-structures and Lie derivatives

@article{Godina2003ReductiveGA, title={Reductive G-structures and Lie derivatives}, author={Marco Godina and Paolo Matteucci}, journal={Journal of Geometry and Physics}, year={2003}, volume={47}, pages={66-86} }

Abstract Reductive G -structures on a principal bundle Q are considered. It is shown that these structures, i.e. reductive G -subbundles P of Q , admit a canonical decomposition of the pull-back vector bundle i P ∗ (TQ)≡P× Q TQ over P . For classical G -structures, i.e. reductive G -subbundles of the linear frame bundle, such a decomposition defines an infinitesimal canonical lift. This lift extends to a prolongation Γ -structure on P . In this general geometric framework the theory of Lie… Expand

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#### References

SHOWING 1-10 OF 18 REFERENCES

Split structures in general relativity and the Kaluza–Klein theories

- Mathematics, Physics
- 1999

We construct a general approach to the decomposition of the tangent bundle of pseudo-Riemannian manifolds into direct sums of subbundles, and the associated decomposition of geometric objects. An… Expand

Natural operations in differential geometry

- Mathematics
- 1993

I. Manifolds and Lie Groups.- II. Differential Forms.- III. Bundles and Connections.- IV. Jets and Natural Bundles.- V. Finite Order Theorems.- VI. Methods for Finding Natural Operators.- VII.… Expand

Metric-affine gravity and the Nester-Witten 2-form

- Physics
- 2001

In this paper we redefine the well-known metric-affine Hilbert Lagrangian in terms of a spin connection and a spin-tetrad. On applying the Poincare–Cartan method and using the geometry of… Expand

Spineurs, opérateurs de dirac et variations de métriques

- Mathematics
- 1992

In this article a geometric process to compare spinors for different metrics is constructed. It makes possible the extension to spinor fields of a variant of the Lie derivative (called the metric Lie… Expand

A geometric definition of Lie derivative for Spinor Fields

- Physics
- 1996

Relying on the general theory of Lie derivatives a new geometric definition of Lie derivative for general spinor fields is given, more general than Kosmann's one. It is shown that for particular… Expand

Einstein-Dirac theory on gauge-natural bundles

- Physics, Mathematics
- 2002

We present a clear-cut example of the importance of the functorial approach of gauge-natural bundles and the general theory of Lie derivatives for classical field theory, where the sole correct… Expand

Gauge Formalism for General Relativity and Fermionic Matter

- Physics
- 1998

A new formulation for General Relativity is developed; it is a canonical, global and geometrically well posed formalism in which gravity is described using only variables related to spin structures.… Expand

Analysis, manifolds, and physics

- Physics
- 1977

Contents. Preface. Preface to the second edition. Preface. Contents. Conventions. I. Review of fundamental notions of analysis. II. Differential calculus on banach spaces. III. Differentiable… Expand

Dérivées de Lie des spineurs

- Mathematics
- 1971

RésuméOn étudie le problème de la transformation d'un champ de spineurs sur une variété spinorielle par un groupe à un paramètre de difféomorphismes de la variété, en généralisant aux champs de… Expand