Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 729X, F A is the curvature of the connection A, and s ∈ Γ(X, S + ⊗ E). As aproof, in local coordinates (i ≤ j)D 2 As = e i ∇ Ai (e j ∇ Aj s) = e i e j ∇ Ai ∇ Aj s = −∇ A2i s+e i e j (∇ Ai ∇ Aj −∇ Aj ∇ Ai )s,where ∇ A = ∇ ⊗ 1 + 1 ⊗ ˜∇ A . The first summand is ∇ ∗ A∇ A [Roe (1988)],and the second splits into a term which corresponds to the scalar curvatureon X [Roe (1988)] and the curvature -iF A of the connection A (note thathere we identify the Lie algebra of U(1) with iR).4.14.2.2 Spin and Spin c StructuresThe group Spin(n) is the universal covering of SO(n).The group Spin c (n) is defined via the following extension:1 → Z 2 → Spin c (n) → SO(n) × U(1) → 1, (4.211)i.e., Spin c (n) = (Spin(n) × U(1))/Z 2 .The extension (4.211) determines the exact sheaf–cohomology sequence:· · · → H 1 (X; Spin c (n)) → H 1 (X; SO(n)) ⊕ H 1 (X; U(1)) δ → H 2 (X; Z 2 ).(4.212)By the standard fact that H 1 (X; G) represents the equivalence classesof principal G−-bundles over X, we see that the connecting homomorphismof the sequence (4.212) is given byδ : (P SO(n) , P U(1) ) ↦→ w 2 (P SO(n) ) + ¯c 1 (P U(1) ),where ¯c 1 (P U(1) ) is the reduction mod 2 of the first Chern class of the linebundle associated to the principal bundle P U(1) by the standard representationand w 2 is the second Stiefel–Whitney class.A manifold X has a Spin c −structure if the frame bundle lifts to aprincipal Spin c (n) bundle. It has a Spin−structure if it lifts to a Spin(n)principal bundle.From the above considerations on the cohomology sequence (4.212), andanalogous considerations on the group Spin(n), it follows that a manifold Xadmits a Spin c −-structure iff w 2 (X) is the reduction mod 2 of an integralclass. It has a Spin−-structure iff w 2 (X) = 0. Different Spin c −-structureson X are parametrized by 2H 2 (X; Z) ⊕ H 1 (X; Z 2 ). This follows directlyform (4.212): see [Libermann and Marle (1987)].