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138 10 Generation techniquesDescribing a subspace as a map Φ : V l →V k we have seen in Chapter 2that forms transform under the inverse map Φ ∗ ; they can be “pulled back”by Φ ∗ onto V l where we denote them by ˜ω A and d˜ω A . Since the contractionof ω A with any vector in V l and dω A with any pair of vectors in V l bothvanish, the condition determining integral submanifolds can be stated as˜ω A = 0 and d˜ω A = 0. For a general set of forms the integral manifold isdefined by requiring them to vanish when pulled back onto it.Since the forms ω A ,dω A , etc. only enter the definition of an integralmanifold through linear homogeneous equations, any algebraically equivalentset can be used. Indeed it is the entire ideal – defined as in e.g.Cartan (1945) – of forms generated by the given forms, here ω A and dω A ,which vanishes on an integral manifold. 1-forms like f A ω A , 2-forms likeψ A ∧ ω A + f A dω A (with arbitrary functions f A and 1-forms ψ A ) etc. obviouslyall vanish when pulled back to the integral manifold. An ideal offorms is called closed if it contains the exterior derivatives of all forms inthe set; we shall consider only closed ideals.Introducing coordinates y a in the immersing space V k and x α in thesubspace V l we get with ω A = ω A a dy a and dω A = ω A a,b dyb ∧dy a˜ω A = ωaA ∂y a∂x α dxα , d˜ω A = ωa,bA ∂y b ∂y a∂x β ∂x α dxβ ∧ dx α . (10.39)Setting these expressions equal to zero gives a coupled set of partial differentialequations for k unknown functions y a in terms of l independentvariables x α .How can the integral manifold of a given set of forms be constructed,at least in principle? To this end we consider some general point p withcoordinates y a and a vector V 1 such thatV 1 ω A = 0 (10.40)at p. These are homogeneous linear algebraic equations for the componentsof V 1 in terms of the components of ω A . Denote the rank of thissystem by s 0 ;thusl 1 = k − s 0 components of V 1 can be chosen arbitrarily.We assume the rank of (10.40) to be maximal at the point p and ina neighbourhood thereof. We now construct a one-dimensional integralmanifold V 1 = {y a | y a = f a (s)} by integratingdf a /ds = V a 1(s). (10.41)In doing so we have made l 1 choices of functions of one variable. At eachpoint of V 1 we now choose a vector V 2 such thatV 2 ω A =0, V 1 V 2 dω A =0. (10.42)