Flow of a metric

Definition

Definition with symbols

Let $M$ be a Riemannian manifold. A flow of a metric on $M$ is defined as a map from an interval (possibly infinite) in the real numbers, to the space of all possible Riemannian metrics on the manifold. The domain of this map is often called the time domain and we think of the Riemannian metric as evolving with time. We typically require the flow to be smooth, that is, the metric should vary smoothly as a function of time.

Notions

Invariants of a flow

We may require that the flow does not change a certain property, or quantity, associated with the metric. For instance, a conformal flow is a flow that does not change the conformal class of the metric (viz the metrics at any two points in time are conformally equivalent). A volume-normalized flow does not change the total volume of the manifold (viz at any two points in time, the total volume of the manifolds is the same).

Goal of the flow

The goal of the flow may be to eventually reach a nice metric, such as:

• A constant-curvature metric
• An Einstein metric

and so on

A flow is said to terminate in finite time if it reaches its goal in finite time, that is, after a finite time, the metric stops evolving. A flow is said to converge if, in the limit, it reaches a particular kind of flow.