Lagrangian Submanifolds In Complex Space Forms

Sympletic manifolds and their Lagrangian submanifolds are very important researchobjects in diferential geometry, they play an important role in mathematical physics, andespecially in Mirror symmetry theory and string theory. In this thesis, we study system-atically the classification and characterization problems of the Lagrangian submanifoldsin complex space forms for which the second fundamental form satisfies some geometricproperties. The main results are the following four parts:Firstly, we completely classify the isotropic Lagrangian submanifolds in complexEuclidean space and complex hyperbolic space. We get the expressions for these twokinds of submanifolds for the first time by the method of parametrization, and hence weobtain some new examples.Secondly, we introduce the notion of submanifolds with isotropic cubic tensor, andwe show that Whitney spheres in complex space forms and the isotropic Lagrangiansubmanifolds in complex space forms have isotropic cubic tensor, hence we get a newproperty for them. At the same time, we give a complete classification of the Lagrangiansubmanifolds in3-dimensional complex space forms with isotropic cubic tensor.Next, we study the Calabi product Lagrangian immersions in complex projectivespace and complex hyperbolic space. Starting from two lower dimensional Lagrangianimmersions with certain special geometric property and a Legendre curve, one can con-struct a new Lagrangian immersion, which has the same special geometric property. Weshow that a given Lagrangian submanifold can be obtained as a Calabi product if andonly if its second fundamental form has some special form.At last, we study the Lagrangian submanifolds of complex projective space withparallel second fundamental form. H. Naitoh studied such kind of submanifolds in a se-ries of papers, using the theory of symmetric spaces and Lie groups, and whereas in theirreducible case the classification is clear, little information is given on how to constructall reducible examples. In this thesis, we use an elementary geometric method to studythis kind of submanifolds. As one of the main results of this thesis, we prove that such aLagrangian submanifold is either totally geodesic or a Calabi product, or one of the fourkinds of examples given by H. Naitoh. Therefore, we answer the question of how to con- struct all reducible examples, and hence give a complete and explicit classification of theLagrangian submanifolds in complex projective space with parallel second fundamentalform.