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On purely real surfaces in Kaehler surfaces

An immersion φ : M → ˜ M 2 of a surface M into a Kaehler surface is called purely real if the complex structure J on ˜ 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 C 2 which satisfy the equality case of the inequality.