Thread: Eikonal equation from Riemanian Length Structure

Given a Riemanian manifold X and a positive scalar velocity field v(X) and the length structure where r(t) is some arbitrary path

then derive the isotropic Eikonal equation for the manifold X and prove that it is equal to


where d(x) is the distance from a source point x0 to x both in X.
and is the length of the minimal geodesic from x0 to x as defined by the riemanian metric and the length structure above.


My problem is that I dont know how to approach this. What constitutes a proof for this?

*bump

I still havnt solved this and although I have written a half hearted almost certainly incorrect solution I am still interested in knowing if someone else can help me in the right direction.

I started off my halfhearted attempt to a solution by saying that the integral simplifies to
integral(sqrt<gamma'(t),gamma'(t)>.dt,0,1)

where gamma'(t) is the derivative of a pointwise velocity scaled r'(t)

however the above integral induces the length structure d(x) which by fermats principle should have the same magnitude as the characteristic along which it lies i.e gamma(t) ?? !

so |grad(d)|=|gamma(t)|=|r(t)/v|=|r|/v=1/v

So we have proved what we set out to prove.

But in all honesty it looks wrong from any way you look at it. Any suggestions would be welcome.