This talk consists of two parts; the first one dealing with a study of certain positive curvatures under the Ricci flow and the second one dealing with a rigidity question in geodesic flows.In 2010 Buchard Wilking proved that for every Ad_SO(n;C)-invariant subset S of the Lie algebraso(n;C) one can attach a notion positive curvature, which we call positive S-curvature, which is preserved by the Ricci flow. We study the properties of positive S-curvatures in reference to the Ricci flow. This part of work is in collaboration with Harish Seshadri and Soma Maity. We shall also discuss a problem motivated by blow-up considerations coming from a conjecture of Richard Schoen.In the second part of the talk we will study a rigidity question in Riemannian geometry, viz, when isRiemannian manifold determined by its geodesic flow? After a brief overview of this problem, We will present our work on the rigidity of the at cylinder. This is a joint work with C. S. Aravinda.
Gururaja H. A.
ON WILKING'S CRITERION FOR THE RICCI FLOW AND A RIGIDITY QUESTION IN RIEMANNIAN GEOMETRY