Calculation of Matrix

Let be a smooth family of matrix. The determinant of is given as

.

We can calculate the derivative of the determinant of a time-dependent matrix as following.

where is the adjoint matrix of $A$.

Let be a -dimensional smooth manifold and be a smooth family of Riemannian metrics on . The corresponding volume form is given as

.

The following differential equation is known as Ricci Flow: where is Ricci curvature of . The Scalar Curvature of is given as .

Then applying above calculation we have

.

Consider the following Normalized Ricci Flow . — ()

For this flow the evolution of the Volume is:

.

So we see the volume is preserved under the normalized Ricci Flow ().

Given a solution for of the Ricci Flow , the metrics where

and

are a solution of the Normalized Ricci Flow with . Hence solutions of the normalized Ricci flow differ from solutions of the Ricci flow only by rescalings in space and time.

Under Ricci Flow,

regions of positive curvature tends to shrink;

regions of negative curvature tends to extend.

Under Normalized Ricci Flow,

regions which has more positive curvature than average tends to shrink;

regions which has more negative curvature than average tends to extend.

Let be the geodesic flow induced by the metric , and be the topological entropy of that geodesic flow.

Canonically there is a Liouville measure on the bundle $SM$ associated to and the associated metric entropy .

Under some assumption,

the entropy is decreasing under the flow (at least for a short time), and the minimum of this enropy among the negative sect. curv. with fixed volume is attained by the constant nega. curv..

For Liouville measure the following behavior is **more possible**, under some assumption:

the entropy is increasing under the flow (at least for a short time), and the maximum of this enropy among the negative sect. curv. with fixed volume is attained by the constant nega. curv..

This is strange.