Categories. Riemannian Manifold Transform

As a result of the applications in the control theory, PDE and pattern recognizition, it makes sense of finding out the geometry characteristics under the transform theory. In this thesis, by the existence of the nonholonomic connection of nonintegral distribution, we prove the existence and uniqueness of the sub-Riemannian connection and extend some results of classical transform to notions of sub-Riemannian manifolds.In chapter 2 we recall the basic facts about Riemannian manifold and some notions of the transform theory on Riemannian manifolds.In chapter 3, for sub-Riemannian manifolds we introduce the concept of nonholonomic connection and Shouten tensor. Furthermore, we give a new proof of the existence and uniqueness of the sub-Riemannian connection.In chapter 4, we extend the classical theory of transform to notions of sub-Riemannian manifold, and obtain some invariants. Compared with Riemannian manifold, we can also obtain that these transform invariants are coherence to Riemannian manifold, when the distribution is integral.