On the energy-momentum tensor

In this thesis, the lower bounds which were obtained by O. Hijazi for the eigenvalues of the Dirac operator constructed on spin manifolds are examined. Energy-Momentum tensor, which is defined on the basis of spinor bundle built on Spin manifolds, is examined. Accordingly, the trace of the Energy-Momentum tensor is obtained locally. This study consists of five main sections. In the introduction, a brief historical outline of the work on manifold theory and Dirac operators is presented. In the second part, necessary definitions and concepts such as vector fields, 1-forms and tensors are given which are the basis for our thesis. In the third chapter, the definition of vector bundles and principle bundles are given. Afterwards, basic information about how to construct Spin manifold is mentioned. Moreover, spinor bundles are constructed on Spin manifolds. At the end of this section, the definition of Dirac operator is given. In the fourth chapter, the research findings of the lower bound for the eigenvalues of the Dirac operator constructed on the spin manifolds are given and the limiting case of this lower bound is investigated. In the last chapter, since the first subject of the researched subject is the first one in the field, it is discussed how the results will form the basis for the next studies.