The Noncommutative Residue And Gauss-bonnet Theorems

In this thesis,we study the noncommutative residue and Gauss-Bonnet theorems in different spaces.Firstly,we give the geometric environment for studying the noncommutative residue and some basic facts and important theorems.In order to study the Kastler-Kalau-Walze type theorem on manifolds associated with self-adjoint operators,we study the noncommutative residue of the modified Novikov operator,and give the Lichnerowicz formula of the modified Novikov operator.We also prove the Kastler-Kalau-Walze type theorem related to the modified Novikov operator.Similarly,the noncommutative residue of twisted Dirac operators and twisted signature operators are studied.In addition,we give the symbol of the third power of the twisted Dirac operators and twisted signature operators,and the KastlerKalau-Walze type theorems about twisted Dirac operators and twisted signature operators are proved.Secondly,in order to solve the problem on the Gauss-Bonnet theorem in spaces and manifolds with sub-Riemannian structure,we calculate the sub-Riemannian limits of curvature of curves in the affine group and the Minkowski plane of rigid motion group respectively,and study the sub-Riemannian Gaussian curvature of surfaces on affine group and Minkowski plane of rigid motion group and geodesic curvature of curves on surfaces.At the same time,the Gauss-Bonnet theorems on affine groups and Minkowski plane of rigid motion group are also proved respectively.Similarly,the sub-Riemannian Gaussian curvature and geodesic curvature of curves on the surface of twisted Heisenberg group are also studied,and Gauss-Bonnet theorems on BCV spaces and the twisted Heisenberg group are also proved.The structure of this paper is as follows:In chapter 1,we mainly introduce the research background and development of GaussBonnet theorems on different spaces and the noncommutative residue.In addition,the research content and structure arrangement of the whole thesis are given.In chapter 2,we mainly study the noncommutative residue of different special operators.we study the non-commutative residue on manifolds with boundary associated with modified Novikov operators,twisted Dirac operators and twisted signature operators.In chapter 3,we mainly study the Gauss-Bonnet theorem on affine groups,Minkowski plane of rigid motion groups,BCV spaces and twisted Heisenberg groups.