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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.

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