Geometrical Analysis of Differential Operator Theory on Complex Manifolds

Abstract

Abstract In this research we have used the theory of classification of Fiber bundles and Atiyah-Singer index theorem to study the geometrical and topological properties of differential operator on complex manifolds. Of particular interest is Dirac equation and the generalized global form of Laplace equation. We have also treated the index of De Rham complex that generalizes the Gauss-Bonnet theorem. Our point of view in this treatment is that the De Rham complex represents only one possibility of other complexes that describe differential operators. These geometrical complexes include spin complex which give the moduli space of Dirac equation. All these concepts are related by the topological index of the operator. .