(1) completely solved an open problem raised by P. Malliavin, an academician of French Academy of Sciences, and proved the global existence and uniqueness of geodesics connected by Markov and the quasi-invariance of Wiener measure in path space. (2) We prove the compactness of all (r, p)- capacities in path space and the non-isomorphism of Ito maps in Dirichlet sense. This achievement was quoted twice by Professor D. Elworthy, a senior British stochastic differential geometer, in the 45-minute invitation report of the 2006 Madrid International Congress of Mathematicians. (3) The Lp boundedness of Riesz transform is established on noncompact complete Riemannian manifold with Ricci curvature satisfying certain integrable conditions, which breaks through the strict limitation of the boundedness under Ricci curvature uniformity in previous literature. This achievement has been concerned and cited by many experts in the field of harmonic analysis on noncompact manifolds. Under the condition that Bakry-Emery Ricci curvature is bounded, the martingale expression formula of Riesz transform is proved, and the best asymptotic estimation of Lp- norm of Riesz transform is obtained. (4) Under the condition of the best Bakry-Emery Ricci curvature dimension, the Liouville theorem of symmetric diffusion operators on noncompact complete Riemannian manifolds and the uniqueness of solutions of thermal equations are established. In cooperation with others, Cheeger-Gromoll splitting theorem is proved under appropriate Bakry-Emery Ricci curvature conditions. The latter result improves the work of A. Lichnerowicz, an academician of French Academy of Sciences. (5) Under suitable Weitzenbock curvature conditions, the Lp-Hodge decomposition theorem and Lp- cohomology disappearance theorem on noncompact complete Riemann and Kahler manifolds are established, the existence of Lp- solutions of De Rham equation and Cauchy-Riemann equation is proved, and the Lp- estimation of solutions is established.