Complete noncompact Riemannian manifolds with nonnegative sectional curvature arise naturally in the Ricci flow when one takes the limits of dilations about a singularity of a solution of the Ricci flow on a compact 3-manifold [ H-95a]. To analyze the singularities in the Ricci flow one needs to understand these manifolds in depth. There are three invariants, asymptotic scalar curvature ratio, ...

We derive matter collineations for some static spherically symmetric spacetimes and compare the results with Killing, Ricci and Curvature symmetries. We conclude that matter and Ricci collineations are not, in general, the same.

i ii Chapter 1 Introduction In [11], Hamilton determined a sharp differential Harnack inequality of Li–Yau type for complete solutions of the Ricci flow with non-negative curvature operator. This Li–Yau–Hamilton inequality (abbreviated as LYH inequality below) is of critical importance to the understanding of singularities of the Ricci flow, as is evident from its numerous applications in [10],...