| Institution: | Department of Mathematics, College of Natural Science, Hanyang University, Sungdong-gu, Seoul 133-791, Korea ; Department of Mathematics, College of Natural Science, Inha University, Incheon-si 402-751, Korea ; Department of Mathematics, College of Natural Science, Dongguk University, Joong-gu, Seoul 100-715, Korea ; Department of Mathematics Education, College of Education, Hongik University, Mapo-gu, Seoul 121-791, Korea |
| Abstract: | We show that, on an oriented Riemannian 4-manifold, existence of a non-zero parallel spinor with respect to a spin structure implies that the underlying smooth manifold admits a Kähler structure. A similar but weaker condition is obtained for the 4-manifold to admit a symplectic structure. We also show that the  structure in which the non-zero parallel spinor lives is equivalent to the canonical spin structure associated to the Kähler structure. |
