Abstract: the self-adjointness of the dirac operator on closed manifold is well-known. While this property is a little bit sutble on a conical manifold. It turns out only the conical manifold with small cone angle capture the self-adjointness of the dirac operator. The existence of harmonic spinor with integrable L^2 norm prevents the dirac operator to be self-adjoint. Based on this observation due to Chou, we prove some interesting properties on conical  manifold.