Given a manifold $M$ with two Riemannian metrics $g_1$ and $g_2$ on $M$, a vector field $\xi$, and a smooth function $f:M\to \mathbb{R}$, is this possible to calculate $g_2(\xi,\xi)$ having the following information:
$g_2(\nabla f,\xi)=constant$, along each $f^{-1}(c)$,
$g_2(\nabla f,\nabla f)=constant$, along each $f^{-1}(c)$,
$\xi$ is orthogonal to $f^{-1}(c)$ with respect to $g_1$ and $\nabla f$ is orthogonal to $f^{-1}(c)$ with respect to $g_2$, ,
$g_1(\xi,\xi)=1$.
Here $c$ is not a critical value of $f$.
P.S. We know nothing about the angle between $\xi$ and $\nabla f$. Here $\nabla f$ is the gradient of $f$ with respect to $g_2$.
Even if one can show that $g_2(\xi,\xi)=constant$, it will be so helpful.