  Cartan-Hadamard manifold is a simply connected Riemannian manifold with
non-positive sectional curvature. In this article, we have proved that a
Cartan-Hadamard manifold satisfying steady gradient Ricci soliton with the
integral condition of potential function is isometric to the Euclidean space.
Next we have proved a compactness theorem for gradient shrinking Ricci soliton
satisfying some scalar curvature condition. Finally, we have showed that a
gradient expanding Ricci soliton with linear volume growth and positive
potential function is an Einstein manifold.
