Geometrical Analysis In Sub-riemannian Manifolds Associated With A Semi-symmetric Connection

A sub-Riemannian manifold(M,?,g)is a smooth differential manifold equipped with a sub-Riemannian structure(?,g),where ?(?)TM is a linear sub-bundle and g is a Riemannian metric defined on ?.If? = TM,then the sub-Riemannian manifold(M,?,g)will degenerate to be a Riemannian manifold(M,g).One of essential difference between the Riemannian ge-ometry and sub-Riemannian geometry is that,there exist a class of minimal singular geodesics,which indicates the importance and difficulty to study the sub-Riemannian geometry.Sub-Riemannian manifolds relate closely to the theory of geometric control,CR manifold,image processing and non-holonomic mechanical system.There are a lot of investigations on sub-Riemannian manifolds,such as analysis,PDE,algebra and geometry recently.Many inter-esting results are obtained.On the other hand,since the energy momentum tensor of an ideal fluid in fluid mechanics is a special case of the Ricci-curvature tensor,and the Black-Scholes equation in finance can be transformed into a semi-harmonic equation,the semi-symmetric con-nections are used widly.Therefore,it is of good theoretical and practical significance to study the geometry and analysis of sub-Riemannian manifold based on the semi-symmetric connec-tion.Firstly,we introduce the definition of semi-symmetric metric connection on sub-Riemannian manifold,derive the corresponding geometric invariants;By using these invariants,we give a characteristic of sub-Riemannian manifold admitting semi-symmetric connection to be flat;Then we study a class of projective transformation which keep the non-holonomic geodesics invariant,and discuss the geometric characteristic of sub-Riemannian manifold admitting semi-symmetric projective connection.As the generation of semi-symmetric metric connection,we discuss the quater-symmetric projective transformations in Riemannian space.Some invariants are obtainedSecondly,we define the semi-harmonic function over sub-Riemannian manifolds.The relations among semi-harmonic functions,harmonic functions and sub-harmonic functions are discussed in Heisenberg group.We obtain some characterizations of the sub-Riemannian geodesics based on the semi-symmetric metric connection.Again,we define the semi-symmetric non-metric connection in nearly Riemannian man-ifold based on the nearly Riemannian connection.The corresponding geometric invariants and geometric and physical characteristics of sub-Riemannian manifold admitting the semi-symmetric non-metric connection are obtained.Finally,we study a class of asymptotically sub-Riemannian Heisenberg manifolds,define the ADM mass based on symmetric and semi-symmetric metric connection.The corresponding positive mass theorem are proved.