where qT is a complex-valued one-form, defined up to a factor of absolute value 1. This form qp defines a complex structure on M. For x E M the space f(x) has an orthogonal space f(x) ' of dimension n - k. We denote by [ f(x)] and [ f(x) '] their corresponding projective spaces, of dimensions k - 1 and n - k - 1,
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Chern et al. (1987) studied this question.