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20 2 Differential geometry without a metricFrobenius’s theorem is important in the construction of exact solutionsbecause it allows us to introduce local coordinates f, h adapted to givennormal 1-forms (see e.g. §27.1.1).The rank q of a 2-form α is defined byα ∧···∧α} {{ }q factors̸= 0, α} ∧···∧α {{ } =0, 2q ≤ n. (2.46)(q+1) factorsUsing this definition we can generalize the statement (2.44) toTheorem 2.3 (Darboux’s theorem). Let σ be a 1-form and let the 2-formdσ have rank q. Then we can find local coordinates x 1 ,...,x q , ξ 1 ,...,ξ n−qsuch that{ =0:σ = x 1 dξ 1 + ···+ x q dξ q ,if σ ∧ dσ } ∧···∧dσ {{ } ̸= 0:σ = x 1 dξ 1 + ···+ x q dξ q +dξ q+1 (2.47).q factors(For a proof, see Sternberg (1964).)This theorem gives the possible normal forms of a 1-form σ. SpecializingDarboux’s theorem to a four-dimensional manifold one obtains thefollowing classification of a 1-form σ in terms of its components:q =0:σ [a,b] =0: σ a = ξ ,aq =1:σ [a,b] ̸= 0, σ [a,b σ c,d] =0,σ [a,b σ c] =0:σ a = xξ ,aσ [a,b] ̸= 0, σ [a,b σ c,d] =0,σ [a,b σ c] ̸= 0:σ a = xξ ,a + η ,a(2.48)q =2:σ [a,b σ c,d] ̸= 0 σ a = xξ ,a + yη ,a .The real functions denoted by x, y, ξ, η are independent. The secondsubcase is just Frobenius’s theorem applied to a single 1-form σ.Now we give a theorem concerning 2-forms.Theorem 2.4 For any 2-form α of rank q there exists a basis {ω a } suchthatα =(ω 1 ∧ ω 2 )+(ω 3 ∧ ω 4 )+···+(ω 2q−1 ∧ ω 2q ). (2.49)If dα =0, then we can introduce local coordinates x 1 ,...,x q , ξ 1 ,...,ξ n−qsuch thatα =dx 1 ∧ dξ 1 + ···+dx q ∧ dξ q . (2.50)(For a proof, see Sternberg (1964).)To conclude this series of theorems, we consider a map Φ : M→Nbetween two manifolds, as in (2.23), and show by induction