Connections of Differential Operators
摘要
A connection 0 of a vector bundle F rnay be regarded to be the lower order term of a differential operator D : Cco(M, APT"(M)(E9F)--Co"(M, AP"'T"(M)(E3>F) with the symbol ff (d)opidF (cf.[1]).Similarly, for an arbitrary differential operator D:Coo(M, Ei)--,Coo(M, E2), Ei and E2 being vector bundles over M, we may consider the Iower order term of a differential operator D : Coo(M, Ei(g)F)-Coe(M, E, <EbF) with a(D)==a(D)opidF, (a(D), etc., mean the symbols of D, etc.), to be a connection of D with respect to F. This connection has many (formally) similar properties as usual connection.For example, the action of the group of automorphisms of F to the set of all connections of D with respect to F is formally same as usual case (cf.[9]), and the obstruction class o(D, F) which has similar properties as curvature or characteristic classes, can be defined by the help of connec-The outline of this paper is as follows:In g1, we define the connection of D with respect to a vector bundle F. After showing the existence of connection, the action of the automorphism of F to the connection is calculated in S1.In g2, we define the obstruction class o(D, F) and show D has a connection with respect to F with the degree at most degD-2 if and only if o(D, F)=O, The higher obstructions oi(D, F) are also defined under the assumption oi-i(D, F)::=O.It is shown that D has a connection with tlie degree at most degD-1'-1 if and only if oi(D, F)=O.If Fisa complex line bundle, o(d, F)EH'(M, uei), uei is the sheaf of germs of closed 1-forms on M, and its de Rham image in H2(M, C) is the 1-st Chern ciass of F, the closed 2-form on Mwhose de Rham image o(d.F), is the curvature form of F. For this reason, we may define ch(D, F) and chY(D, F) using non-abelian cohomogy theory ([6], [8]).In g3, we consider the extension of differential operator D on the base space M to the tatal space MF of a fibre bundle F and show this problem is also treated by the same way as the connection of D defined in S1.Fo