TORSION TENSOR FORMS ON INDUCED BUNDLES

Abstract

Abstract. Let ` be a map of a manifold M into another manifold N , L ( N ) the bundle of all linear frames over N , and ` i 1 ( L ( N ))the bundle over M which is induced from ` and L ( N ). Then, weconstruct a structure equation for the torsion form in ` i 1 ( L ( N ))which is induced from a torsion form in L ( N ). 1. IntroductionLet ` : M ! N be a C 1 i map between smooth manifolds M and N , L ( N ) the bundle of all linear frames over N , and ` i 1 ( L ( N )) =: Q the bundle which is induced from ` and L ( N ). Then, the bundlehomomorphism `~ : Q ! L ( N ) between two principal flbre bundles ` i 1 ( TN ) =: Q and L ( N ) is deflned by `~ ( u;x ) = u (( u;x ) 2 Q;x 2M;… ( u ) = x;u 2 L ( N )) ([1]).Let i be an arbitrarily given connection in L ( N ). Let ! and µ be theconnection form and the canonical form in L ( N ) which are deflned fromi ([1, 3]). Now, putting `~ ⁄ ! =: ~ ! and `~ ⁄ µ =: µ~ , we obtain the fact that !~ and µ~ are a connection form and the canonical form for ~ ! in `