Russian Mathematics Iz VUZ Izvestiya VUZ Matematika Vol No pp UDC SPINORS ON THE RIEMANNIAN MANIFOLDS RF Bilyalov Intro duction In the most general mathematical form the spinors were intro duced by E Cartan in see In the spinors were discovered again byvan der Waerden see in connection with Diracs physical investigations VA Fock and DD Ivanenko implemented the spinors into the general relativity see In L Rosenfeld suggested a general pro cedure for constructing the energy momentum tensor of material elds which stands in the righthand side of Einsteins equations see L Rosenfelds metho d essentially involves Lie derivative Construction of Lie derivative for tensor elds encounters no diculties but this is not true for spinor elds In the deduction of spinor elds L Rosenfeld supp osed that a Lie of a formula for the energymomentum tensor derivativeof spinor eld exists suchthat the partial derivativecommutes with the Lie derivative A Lie derivative of spinor eld with resp ect to a Killing vector eld had b een rst constructed by A Lichnerowicz see and then Y Kosmann p ostulated a formula for Lie derivative of spinor by analogy with the Lie derivative of tensor eld see An explanation for this formula was given in However Y Kosmans version of Lie derivative has not prop erty that the commutator of Lie derivatives is the Lie derivative with resp ect to commutator to say nothing of the prop erty that the partial derivativecommutes with the Lie derivative The problem on constructing the Lie derivative of a spinor eld was solved in where this Lie derivativewas used to construct the energymomentum tensor of spinor elds in the space time of general relativity on the basis of No ethers theorem In the metho d of induced representations was used in order to expand the spinor representation of the Lorentz group O to a representation of the general linear group GL of fourdimensional space The space of representation is also expanded this one is the spinor space multiplied by the space of symmetric forms of typ e which determine the quadratic form of signature With the use of the constructed representation the spinors which earlier were treated as elements of a bundle asso ciated with the principal bundle of orthonormal frames now are considered as elements of a bundle asso ciated with the principal bundle of linear frames In this article we generalize results from to arbitrary Riemannian manifolds We demon strate that our construction of spinors on a Riemannian manifold dep ends essentially on a choice of sections of a principal bundle with total space GLn and structure group O p q n p q In addition we prove that if the Lagrange function of spinor eld is invariant with resp ect to the action of structure group on the asso ciated spinor bundle then the corresp onding theory of spinor eld is gauge invariantincase the gauge transformation is a change of section c by Allerton Press Inc Authorization to photo copy individual items for internal or p ersonal use or the internal or p ersonal use of sp ecic clients is granted by Allerton Press Inc for libraries and other users registered with the Copyright Clearance Center CCC Transactional Rep orting Service provided that the base fee of p er copy is paid directly to CCC Rosewo o d Drive Danvers MA .