Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

(propose a free ad)

Let $(M,g)$ be a riemannian manifold with riemannian curvature $R$.

$$r_g(x,y,z,t)=g(R(x,y)z,t)$$

The Riemann flow for the metric $g$ is defined by the following equation:

$$\frac{\partial}{\partial t}[g(x,x)g(y,y)-g(x,y)^2]=r_g(x,y,x,y)$$

The definition is coherent because if $x=y$, the result is zero.

Is the Riemann flow really well defined and has solutions?

It looks like x,y,z,t are arbitrary vector fields here, and $$\frac{\partial}{\partial t} $$ is the Lie derivative along t. You're asking about the existence of a g that satisfies the equation for all vector fields x,y,z,t. Is all that correct?

If that's what you mean, then I think this has only trivial solutions. If we replace t with -t, then the sign of the left-hand side of the second equation flips, while the right-hand side stays the same. So the only solution is when both sides are zero.

However, based on the way you chose to define $$r_g$$, I suspect I may not be interpreting the question correctly. Clarification could be helpful.

user contributions licensed under cc by-sa 3.0 with attribution required