2005/04/18 by Marco Godina, Paolo Matteucci · 1 citation
Mathematics · Physics and Astronomy · #math.DG #math-ph #math.MP #msc:53A55 #msc:53C10 #msc:53C27 #msc:58A20
paper · pdf · doi:10.1142/s0219887805000624
published as Int. J. Geom. Methods Mod. Phys. 2 (2005) 159-188 · 28 pages, 1 figure
arxiv created 2005/04/18 · arxiv updated 2009/12/01
Starting from the general concept of a Lie derivative of an arbitrary differentiable map, we develop a systematic theory of Lie differentiation in the framework of reductive G-structures P on a principal bundle Q. It is shown that these structures admit a canonical decomposition of the pull-back vector bundle iP^*(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 Gamma-structure on P. In this general geometric framework the concept of a Lie derivative of spinor fields is reviewed. On specializing to the case of the Kosmann lift, we recover Kosmann's original definition. We also show that in the case of a reductive G-structure one can introduce a "reductive Lie derivative" with respect to a certain class of generalized infinitesimal automorphisms, and, as an interesting by-product, prove a result due to Bourguignon and Gauduchon in a more general manner. Next, we give a new characterization as well as a generalization of the Killing equation, and propose a geometric reinterpretation of Penrose's Lie derivative of "spinor fields". Finally, we present an important application of the theory of the Lie derivative of spinor fields to the calculus of variations.