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380 P. Bressler, A. Gorokhovsky, R. Nest, and B. Tsygandefines an isomorphism of sheaves of groupsexpW J M;0 ! Aut 0 .L ˝ J M /;where Aut 0 .L ˝ J M / is the sheaf of groups of (locally defined) J M -linear automorphismsof L ˝ J M making the diagramL ˝ J M L ˝ J MId˝pId˝p Lcommutative.Lemma 7.2. The sheaf Isom 0 .L ˝ J M ; J.L// is a torsor under the sheaf of groupsexp J M;0 .Proof. Since L is locally trivial, both J.L/ and L˝J M are locally isomorphic to J M .Therefore the sheaf Isom 0 .L ˝ J M ; J.L// is locally non-empty, hence a torsor.Corollary 7.3. The torsor Isom 0 .L ˝ J M ; J.L// is trivial, that is, Isom 0 .L ˝ J M ;J.L// WD .MI Isom 0 .L ˝ J M ; J.L/// ¤ ¿.Proof. Since the sheaf of groups J M;0 is soft we have H 1 .M; J M;0 / D 0 (see [8],Lemme 22, cf. also [6], Proposition 4.1.7). Therefore every J M;0 -torsor is trivial.Corollary 7.4. The set Isom 0 .L ˝ J M ; J.L// is an affine space with the underlyingvector space .MI J M;0 /.Let L 1 and L 2 be two line bundles, and f W L 1 ! L 2 an isomorphism. Thenf induces a map Isom 0 .L 2 ˝ J M ; J.L 2 // ! Isom 0 .L 1 ˝ J M ; J.L 1 // which wedenote by Ad f :Ad f./D .j 1 .f // 1 ı ı .f ˝ Id/:Let L be a line bundle on M and f W N ! M is a smooth map. Then there is apull-back map f W Isom 0 .L ˝ J M ; J.L// ! Isom 0 .f L ˝ J N ; J.f L//.If L 1 , L 2 are two line bundles, and i 2 Isom 0 .L i ˝ J M ; J.L i //, i D 1; 2. Thenwe denote by 1 ˝ 2 the induced element of Isom 0 ..L 1 ˝ L 2 / ˝ J M ; J.L 1 ˝ L 2 //.For a line bundle L let L be its dual. Then given 2 Isom 0 .L ˝ J M ; J.L//there exists a unique a unique 2 Isom 0 .L ˝ J M ; J.L // such that ˝ D Id.For any bundle E J.E/ has a canonical flat connection which we denote by r can .A choice of 2 Isom 0 .L ˝ J M ; J.L// induces the flat connection 1 ır canLı onL ˝ J M .Let r be a connection on L with the curvature . It gives rise to the connectionr˝Id C Id ˝r can on L ˝ J M .