Journal of Geometry and Physics 53 (2005) 428–460 Classification of Lagrangian surfaces of constant curvature in complex projective plane Bang-Yen Chen ∗ Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027, USA Received 7 April 2004; received in revised form 22 July 2004; accepted 26 July 2004 Available online 13 September 2004 Abstract From Riemannian geometric point of view, one of the most fundamental problems in the study of Lagrangian submanifolds is the classification of Lagrangian immersions of real space forms into complex space forms. The purpose of this article is thus to classify Lagrangian surfaces of constant curvature in complex projective plane CP 2 . Our main result states that there are 29 families of Lagrangian surfaces of constant curvature in CP 2 . Twenty-two of the 29 families are constructed via Legendre curves. Conversely, Lagrangian surfaces of constant curvature in CP 2 are obtained from the 29 families. As an immediate by-product, many interesting new examples of Lagrangian surfaces of constant curvature in CP 2 are discovered. © 2004 Elsevier B.V. All rights reserved. MSC: 53D12; 53C40; 53C42SC JGSSC: Differential geometry Keywords: Lagrangian surface; Legendre curve; Special Legendre curve; Surface of constant curvature 1. Introduction A submanifold M of a Kaehler manifold ˜ M is called Lagrangian if the almost complex structure J of ˜ M interchanges each tangent space of M with its corresponding normal space. ∗ Tel.: +1 517 3534670; fax: +1 517 4321532. E-mailaddress: bychen@math.msu.edu (B.-Y. Chen). 0393-0440/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.geomphys.2004.07.009