Suppose that $(M,g)$ is a compact Riemannian manifold and $f:M\to [a,b]$ a smooth function such that $\|\nabla f\|$ is constant along each level set.
Assume that $\forall c,d \in [a,b]$ and $\forall p \in f^{-1}(c)$ and $\forall q \in f^{-1}(d)$, $$d(p,f^{-1}(d))=d(f^{-1}(c),q).$$ Also we know that $a$ and $b$ are the only critical values and each of the level sets $f^{-1}(a)$ and $f^{-1}(b)$ has dimension less than $n-1$, where $n=\dim M$. Here $\nabla f$ is the gradient of $f$, that is defined as $df_p(v)=g(\nabla f, v)$, $\forall v \in T_pM$.
Is that possible to show that $f^{-1}(a)$ and $f^{-1}(b)$ are (immersed) submanifolds of $M$, with these hypothesis?
One can also prove that each level set $f^{-1}(c)$, where $c\in (a,b)$, is am embedded submanifold of dimension $n-1$.
Also $\forall q \in f^{-1}(a)$ there exists a geodesic that minimizes the distance from $q$ to $f^{-1}(c)$ and this geodesic is orthogonal to each level set $f^{-1}(c)$ and eventually meets $f^{-1}(b)$. One has the same results about $\forall q \in f^{-1}(b)$.