A Non Tight Three-dimensional Complex Manifolds On Complex Coordinate And Ricci Curvature Calculation

We first introduce almost complex structure, hermitian metric and curvature, and a method proposed by Calabi to construct complex three dimensional manifolds. Following Calabi’s method, one can construct a complex manifold M from Catenoid. In this thesis, we find complex coordinates in M, and get an expression of the hermitian metric which deduced by Calabi’s method. Finally, we prove that the Ricci curvature of M is zero.