, By Proposition 7.3.5 and Proposition 7.3.1, M admits a complex structure J such that (M, J, g) is a Hermitian manifold. Now, if (M, J, g) is a Hermitian manifold, by Proposition 7.3.5, the canonical Spin c structure on M admits a pure spinor field ? ? ?(? 0 M ) By Proposition 7.3.1 and since J is a complex structure, the pure spinor field is integrable. Hence, the first two statements are equivalent. The last two statements are also equivalent because if (M, J, g) is a Hermitian manifold with a complex structure J, the eigenbundle T 1,0 M of T M ? C corresponding to the eigenvalue i of J gives the desired almost CR-structure of type ( n 2 = m, 0) Because T 1,0 M is integrable, the almost CR-structure is a CR-structure of type (m, 0) Conversely, a Riemannian manifold having a CR-structure of type (m, 0) is a complex manifold, p.121
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