黎曼Spin流形上的Dirac算子是作用在旋量丛上的一阶椭圆微分算子，该算子与数学物理、偏微分方程等数学诸多领域的研究都有十分密切的联系。二维流形上的一特殊旋量场的存在性意味着该黎曼流形可局部等距浸入到三维欧式空间中。Dirac算子的谱包含了流形的几何和拓扑的微妙信息。本课题的主要任务是利用旋量诱导的能量动量张量去研究超曲面的旋量表示，以及与之相关的Dirac算子特征值的估计问题。

The Dirac operator on a Spin manifold is a first order elliptic differential operator, which acts on the spinor bundle. .The operator has close connection with mathematical physics, partial differential equations and other.fields of mathematics. The existence of a special spinor field--a solution of the Dirac equation on Riemannian surfaces .implies surfaces can be locally isometrically immersed into the 3-dimension Euclidean space. The spectrum of the Dirac .operator on closed spin manifolds detects subtle information on the geometry and the topology of such manifolds. .The main task of this project is, by using the spinorial energy-momentum tensor, to study the spinorial characterizations .of hypersurfaces and related problems about eigenvalue estimates for the Dirac operator.