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Maslovian Lagrangian surfaces of constant curvature in complex projective or complex hyperbolic planes
Author(s) -
Chen BangYen
Publication year - 2005
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200310305
Subject(s) - mathematics , constant curvature , curvature , mean curvature , geodesic , mathematical analysis , constant (computer programming) , vector field , lagrangian , submanifold , killing vector field , principal curvature , surface (topology) , geometry , unit vector , pure mathematics , computer science , programming language
A Lagrangian submanifold is called Maslovian if its mean curvature vector H is nowhere zero and its Maslov vector field JH is a principal direction of A H . In this article we classify Maslovian Lagrangian surfaces of constant curvature in complex projective plane CP 2 as well as in complex hyperbolic plane CH 2 . We prove that there exist 14 families of Maslovian Lagrangian surfaces of constant curvature in CP 2 and 41 families in CH 2 . All of the Lagrangian surfaces of constant curvature obtained from these families admit a unit length Killing vector field whose integral curves are geodesics of the Lagrangian surfaces. Conversely, locally (in a neighborhood of each point belonging to an open dense subset) every Maslovian Lagrangian surface of constant curvature in CP 2 or in CH 2 is a surface obtained from these 55 families. As an immediate by‐product, we provide new methods to construct explicitly many new examples of Lagrangian surfaces of constant curvature in complex projective and complex hyperbolic planes which admit a unit length Killing vector field. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)