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48 2 Complex Manifolds Clearly φ is then a submersion, and each fibre φ −1 (v) is a closed submanifold of U having the property that its tangent space at each point is equal to the fibre of E at that point. The following theorem characterises the integrable distributions. 20 (Frobenius) A distribution E is integrable if and only if for all C 1 vector fields χ , ψ contained in E, the bracket [χ, ψ] is also contained in E. Proof Obviously, if E is integrable, then E is stable under the bracket. e. 19.

Another essential application is the principle of analytic continuation. 22 Let U be a connected open set of Cn , and f a holomorphic function on U . If f vanishes on an open set of U , then f is identically zero. e. locally equal to the sum of its Taylor series). We can thus apply the principle of analytic continuation to f . We recall that the latter is shown by noting that if f is analytic, the open set consisting of the points in whose neighbourhood f vanishes is equal to the closed set consisting of the points where f and all its derivatives vanish.

Let 1 ∈ R, 0 < 1 < |z 1 | be such that the closed disk of radius 1 and centre z 1 is contained in the disk {ζ | |ζ | < r1 }. Then the polydisk D 1 := {(ζ1 , . . , ζn )| |ζ1 − z 1| ≤ 1, |z i | ≤ ri , i ≥ 2} is contained in D − {ζ1 = 0}, so that Cauchy’s formula gives f (z) = 1 2iπ n f (ζ ) ∂D 1 dζ1 dζn ∧ ··· ∧ , ζ1 − z 1 ζn − z n where ∂ D 1 := {(ζ1 , . . , ζn )| |ζ1 − z 1| = 1, |ζi | = ri , i ≥ 2}. 2 Holomorphic functions of several variables 33 Consider, also, the product of circles ∂ D := {(ζ1 , .