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differential geometry w notes from teacher_Part_81

# differential geometry w notes from teacher_Part_81 - are...

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6.3. CARTAN’S STRUCTURAL EQUATIONS 161 Finally, we consider matrix-valued p -forms valued in V V * and extend the operator D to such forms D : Λ p V V * Λ p + 1 V V * by ( D α ) i j = d α i j + A i k α k j - ( - 1) p α i k ∧ A k j or, in matrix form, D α = d α + A ∧ α - ( - 1) p α ∧ A . Now, let σ = ( σ i ), F = ( F i j ) and α = ( α i ) be an arbitrary vector-valued 1-form. Then D σ = 0 D 2 α = F ∧ α DF = 0 . Problem . Let the dimension n = 2 k of the manifold M be even. We deﬁne the following 2 l -forms Ω ( l ) = tr F ∧ ··· ∧ F | ±±±±±±±±± {z ±±±±±±±±± } l and the n -form Ω = ε i 1 ... i 2 k F i 1 i 2 ∧ ··· ∧ F i 2 k - 1 i 2 k 1. Prove that these forms are independent of the orthonormal basis and

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Unformatted text preview: are closed, that is, d Ω ( l ) = d Ω = . 2. Find the expressions in local coordinates for these forms. • These forms deﬁne so called characteristic classes , which are closed in-variant forms whose integrals over the manifold do not depend on the metric and, therefore, are topological invariants of the manifold. di ﬀ geom.tex; April 12, 2006; 17:59; p. 160 162 CHAPTER 6. CONNECTION AND CURVATURE di ﬀ geom.tex; April 12, 2006; 17:59; p. 161...
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