Adiabatic Limits And Vanishing Theorems

In this paper, we construct the sub-Dirac operator, take the adiabatic limit of the square ofthe sub-Dirac operator and compute the adiabatic limit of the scalar curvature,take aLichnerowicz type formula and the Atiyah-Singer index theorem.From those methods, we getthe extended vanishing theorem of A.connes. Taking the similar discussing, take the adiabaticlimit of the square of the sub-Dirac operator ,we get the A.connes type vanishing theoremwhen F isn't integrable, especially for contact manifold.