SPINOR, DIRAC OPERATOR AND CHANGE OF THE METRIC

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 derivative), giving a geometric approach to a construction originally due to Yvette Kosmann. The comparison of spinor fields for two different Riemannian metrics makes the study of the variation of Dirac operators feasible. For this it is crucial to take into account the fact that the bundle in which the sections acted upon by the Dirac operators take their values is changing. We also give the formulas for the change in the eigenvalues of the Dirac operator. We conclude by giving a few cases in which an eigenvalue is stationary.