Reductive /Gstructures and Lie derivatives
Abstract
Reductive Gstructures on a principal bundle Q are considered. It is shown that these structures, i.e. reductive Gsubbundles P of Q, admit a canonical decomposition of the pullback vector bundle i _{P}^{∗}(TQ)≡P× _{Q}TQ over P. For classical Gstructures, i.e. reductive Gsubbundles of the linear frame bundle, such a decomposition defines an infinitesimal canonical lift. This lift extends to a prolongation Γstructure on P. In this general geometric framework the theory of Lie derivatives is considered. Particular emphasis is given to the morphisms which must be taken in order to state what kind of Lie derivative has to be chosen. On specializing the general theory of gaugenatural Lie derivatives of spinor fields to the case of the Kosmann lift, we recover the result originally found by Kosmann. We also show that in the case of a reductive Gstructure one can introduce a "reductive Lie derivative" with respect to a certain class of generalized infinitesimal automorphisms. This differs, in general, from the gaugenatural one, and we conclude by showing that the "metric Lie derivative" introduced by Bourguignon and Gauduchon is in fact a particular kind of reductive rather than gaugenatural Lie derivative.
 Publication:

Journal of Geometry and Physics
 Pub Date:
 July 2003
 DOI:
 10.1016/S03930440(02)001742
 arXiv:
 arXiv:math/0201235
 Bibcode:
 2003JGP....47...66G
 Keywords:

 Mathematics  Differential Geometry;
 Mathematical Physics;
 53A55;
 53C10;
 53C27;
 58A20
 EPrint:
 LaTeX (21 pages, 3 runs)