A Poincaré-hopf Type Formula For The Sections Of A Vector Bundle

In this thesis we will establish a Poincaré-Hopf type formula for a pair of sections of an oriented real vector bundle, which generalized a recent result of Feng-Li-Zhang in [1].This thesis is divided into four chapters. The first chapter is a brief introduction to the background and motivation of our research, also including Poincaré-Hopf formula and its related history. Furthermore, we will present the main results of this thesis. In the second and third chapters, we will introduce some prerequisited materials for this thesis. More precisely, we will give the definitions and examples of Vector bundle, Clifford algebra, Clifford module, superconnection. Also, a brief introduction about Chern-Weil theory of characteristic classes is included. In the last chapter is the mainly research result in this thesis, which gives the proof about a Poincaré-Hopf type formula for a pair of sections of a real vector bundle.