Geometric flows without and with higher order terms

Geometric fl ows have been an active topic of research for the last seven years or so, ever since Perelman's groundbreaking work appeared. In this talk, we begin with a brief overview of the most well-known fl ow, namely, Ricci fl ow. We then move on to our work on Ricci flow of unwarped and warped product manifolds through a study of generic examples. In particular, we look at features such as singularity formation, isotropisation at specic values of the ow parameter and evolution characteristics. Subsequently, we move on to geometric flows with higher order terms such as those involving (Riemann)2. We discuss second or- der (in Riemann curvature) geometric fl ows (un-normalised) on locally homogeneous three manifolds as well as unwarped and warped products of various types. Novelties specically associated with the presence of the higher order terms are pointed out. Finally, we discuss a new geometric ow governed by the Bach tensor, which involves higher derivatives of the metric, as well as higher orders. Here, we rst consider (2; 2) product manifolds where we solve the Bach ow equations for typical unwarped products and also find the fixed point criteria. A brief discussion on Bach flows for warped products is also presented.