6维近凯勒流形中典型子流形的刚性及分类问题研究

发布时间：2019-02-12 11:09

【摘要】：6维近凯勒流形是一类重要的几何对象,对其各种典型子流形的研究是十分自然而重要的课题.本论文研究6维近凯勒流形中的近复曲面和拉格朗日子流形,主要结果如下:(一)得到了齐性近凯勒流形S3×S3中紧致近复曲面的刚性定理.我们对该类曲面建立了Simons型积分不等式,其中的等式成立当且仅当近复曲面是全测地的.这给出了S3×S3中仅有的两类全测地近复曲面的一种新刻画(见定理1.1).并且我们建立的积分不等式给出了Simons积分不等式在外围空间不是局部黎曼对称的,并且曲面余维是4情形下的一个推广.(二)得到了齐性近凯勒流形S3×S3中具有平行第二基本形式的拉格朗日子流形的完全分类.首先我们将近齐性凯勒流形S6中的一个结果推广到一般6维严格近凯勒流形,即证明了6维严格近凯勒流形中具有平行第二基本形式的拉格朗日子流形一定是全测地的(见定理1.2).进一步地,通过研究S3×S3的近乘积结构,我们完全分类了近凯勒流形S3×S3中的全测地拉格朗日子流形(见定理1.3).(三)证明了6维严格近凯勒流形中的迷向拉格朗日子流形一定是全测地的(见定理1.4).这推广了凯勒流形中关于常迷向拉格朗日子流形的一个结果.我们完全分类了齐性近凯勒流形S3×S3中的J-迷向拉格朗日子流形(见定理1.5).该分类定理表明,利用J-迷向条件可以统一刻画S3×S3中所有全测地的和具有常截面曲率的拉格朗日子流形.
[Abstract]:6 dimensional near Keller manifold is a kind of important geometric object, and it is a very natural and important subject to study its typical submanifolds. In this paper, we study the near complex surfaces and Lagrange submanifolds in 6 dimensional near Keller manifolds. The main results are as follows: (1) the rigidity theorem of compact near complex surfaces in homogeneous near Keller manifolds S3 脳 S3 is obtained. We establish an integral inequality of Simons type for this class of surfaces, in which the equation holds if and only if the near complex surface is totally geodesic. This gives a new characterization of the only two classes of total geodesic near complex surfaces in S3 脳 S3 (see Theorem 1.1). Moreover, the integral inequality we establish is a generalization of the Simons integral inequality in the case that the surface codimension is 4, and that the Simons integral inequality is not locally Riemannian symmetric in the peripheral space. (2) the complete classification of Lagrange submanifolds with parallel second fundamental form in homogeneous near Keller manifold S3 脳 S3 is obtained. First, we generalize a result in nearly homogeneous Keller manifold S6 to a general 6-dimensional strictly near Keller manifold. It is proved that the Lagrangian submanifolds with parallel second fundamental form in 6 dimensional strictly near Keller manifolds must be totally geodesic (see Theorem 1.2). Furthermore, by studying the near product structure of S3 脳 S3, we completely classify the total geodesic Lagrange submanifolds in the near Keller manifold S3 脳 S3 (see Theorem 1.3). (3) it is proved that the isotropic Lagrange submanifolds in 6 dimensional strictly near Keller manifolds must be totally geodesic (see Theorem 1.4). This generalizes a result of the constant isotropic Lagrange submanifolds in Keller manifolds. We completely classify the J-isotropic Lagrange submanifolds in homogeneous near Keller manifolds S3 脳 S3 (see Theorem 1.5). The classification theorem shows that all geodesic Lagrange submanifolds with constant sectional curvature in S 3 脳 S 3 can be uniformly characterized by using the J isotropic condition.
【学位授予单位】：郑州大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O186.12

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