Date: Approved:

In this thesis we investigate conditions for the existence of solitons for the Ricci flow. The Ricci flow, first introduced by Richard Hamilton, changes a Riemannian metric over time, in a way that the metric satisfies the partial dif-ferential equation ∂g/∂t = −2Ric(g). “Solitons ” for this flow are solutions of the equation where the metrics at different times differ by a diffeomorphism of the manifold. The soliton condition, sufficient for an initial metric to give rise to a soliton, is LX} = −∈Ric(}). We use the techniques of exterior differential systems to show that this condition is involutive, and gives an elliptic equa-tion in harmonic coordinates. However, globally-defined solitons are harder to obtain on compact manifolds than locally-defined solitons: using techniques from Hamilton’s earlier papers, we show that the only solitons on compact three-manifolds are metrics of constant curvature. For compact manifolds of higher dimension we investigate the possibility of deforming an Einstein met-ric to a soliton, and we calculate the first-order deformation space explicitly in the case of symmetric spaces of compact type. Using warped products and ODE techniques, we also construct some examples of non-compact, complete solitons. i Acknowledgements I would like to thank my advisor, Robert Bryant, for the many hours he spent answering my questions and sharing his insights with me. I am most grateful for his unstinting support during my five years of graduate education. I have also been aided in my dissertation research by advice from Stev