On purely real surfaces in Kaehler surfaces

An immersion Φ: M → tilde M^2 of a surface M into a Kaehler surface is called purely real if the complex structure J on tilde M^2 carries the tangent bundle of M into a transversal bundle. In the first part of this article, we prove that the equation of Ricci is a consequence of the equations of Gauss and Codazzi for purely real surfaces in any Kaehler surface. In the second part, we obtain a necessary condition for a purely real surface in a complex space form to be minimal. Several applications of this condition are provided. In the last part, we establish a general optimal inequality for purely real surfaces in complex space forms. We also obtain three classification theorems for purely real surfaces in C2 which satisfy the equality case of the inequality.