Classification of Lagrangian Surfaces of Curvature ε in Non-flat Lorentzian Complex Space Form
Classification of Lagrangian Surfaces of Curvature ε in Non-flat Lorentzian Complex Space Form
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2009
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English Series
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English Series
在洛伦兹复空间形式<$M2 1（4 e）中，全测地的Lagrange曲面具有常曲率e，这是众所周知的。一个自然的问题是：除了全测地的Lagrange曲面外，在<$M2 1（4 e）中还有多少常曲率e的Lagrange曲面？“在以前的一篇论文中，对这个问题的答案是由Chen和Fastenakels得到的。在本文中，我们提供了这个问题的答案的情况下，e =0。我们的主要结果表明，存在35个家庭的拉格朗日曲面的曲率为e在XM 2 1（4 e），其中e = 0。反之，每一个曲率为e = 0 in <$M2 1（4 e）的拉格朗日曲面局部全等于由35个族给出的拉格朗日曲面之一。
It is well known that a totally geodesic Lagrangian surface in a Lorentzian complex space form ˜ M 2 1(4e) of constant holomorphic sectional curvature 4e is of constant curvature e .A natural question is "Besides totally geodesic ones how many Lagrangian surfaces of constant curvature e in ˜ M 2 1(4e) are there?" In an earlier paper an answer to this question was obtained for the case � =0b y Chen and Fastenakels. In this paper we provide the answer to this question for the case e =0 . Our main result states that there exist thirty-five families of Lagrangian surfaces of curvature e in ˜ M 2 1(4e) with e = 0. Conversely, every Lagrangian surface of curvature e =0i n ˜ M 2 1(4e) is locally congruent to one of the Lagrangian surfaces given by the thirty-five families.