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.
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 $k<n-1$, where $n=\dim M$. Moreover we know that for every $c,d\in [a,b]$, $d(p,f^{-1}(d))=d(f^{-1}(c), q)= d(f^{-1}(c), f^{-1}(d))$, $\forall\ p \in f^{-1}(c)$ and $q \in f^{-1}(q)$.
Here $\nabla f$ is the gradient of $f$, that is defined as $df_p(v)=g(\nabla f, v)$, $\forall v \in T_pM$.
Can one guarantee that there exists an embedded submanifolds $L$ of $M$ of dimension $k$ which is contained in $f^{-1}(b)$? and also the same result for $f^{-1}(a)$?
P.S. By the dimension of $f^{-1}(b)$ we mean the dimension of it is it is a submanifold.