Abstract
In this Dissertation, we study several splitting and vanishing type theorems on some Riemannian manifolds by studying the classes of harmonic functions and of harmonic $p$-forms. First, we prove splitting and vanishing theorem on complete Riemannian manifolds of dimension $n\geq3$ with the Ricci curvature is bounded from below in terms of bottom of spectrum of Laplacian. Moreover, we consider the Riemannian manifolds with weighted Poincar\'{e} type inequality. Assuming a growth condition on the weight function, several splitting theorems and vanishing theorems are proved, moreover some vanishing properties of a special class of $p$-forms are also given on these such spaces. We also consider the smooth metric measure space with Bakry-\'{E}mery curvature. We show a splitting property of smooth metric measure space with the Bakry-\'{E}mery curvature bounded from below in terms of bottom spectrum of weighted Laplacian. Then, we consider a smooth metric measure space of infinite dimension Bakry-\'{E}mery curvature with a weighted Poincar\'{e} inequality. We are successfully in showing very general splitting results in this space. Many works done before now are reproved by them. Last, we mention a special smooth metric measure space, the gradient Ricci soliton. Vanishing type theorems of the class of harmonic $1$-forms which has finite $L^p$-norm are given.