 This correspondence extends straightforwardly to the complex case.Given a pair formula_17 one can construct a maximally isotropic subbundle formula_20 of formula_1 as follows.The elements of the subbundle are the formal sums formula_22 where the vector field "X" is a section of E and the one-form "ξ" restricted to the dual space formula_23 is equal to the one-form formula_24

To see that formula_20 is isotropic, notice that if "Y" is a section of E and formula_26 restricted to formula_23 is formula_28 then formula_29 as the part of formula_26 orthogonal to formula_23 annihilates "Y".