Let $(M,g)$ be a Riemannian manifold and let $N$ be a compact hypersurface isometrically embedded into $M$ and let $\eta$ denote a choice of unit normal vector field on $N$. It is then true that $N$ admits an $\varepsilon$-tubular neighborhood. That is, there exists some $\varepsilon>0$ such that the normal exponential $\exp(t\eta)$ is a diffeomorphism onto its image for $t\in (-\varepsilon, \varepsilon)$. A generalization of the Gauss lemma implies that for each $t$, we have a "parallel" hypersurface $N_t$ which intersects the normal geodesics orthogonally and are distance $t$ from $N$.

I am interested if there has been a result relating the injectivity radius of $N_t$ to the injectivity radius of $N$. More specifically, it seems like it should be the case that the cut locus of $N_t$ varies continuously as one travels along the normal geodesic flow. That is, small change in $t$ should imply small change in the cut time of a point $p(t)$ moving along the normal geodesic. I have not seen a proof of this and to my knowledge the arguments for continuity of the injectivity radius of a manifold do not seem to generalize.

Has this been shown anywhere? The best result that I have found so far is the Riccati equation for tubular neighborhoods in Grays "Tubes" book: $$ S'(t) = S^2(t)+R(t) $$ where $S(t)$ is the shape operator of $N_t$ and $R(t)$ is defined by $R[X,Y] = \nabla_{[X,Y]} - [\nabla_X, \nabla_Y]$ on $N_t$. In this way, it seems like one should be able to relate the second fundamental form of $N_t$ to $N$ via integration. However, since the cut locus is not uniquely determined by the shape operator, this does not seem to be sufficient.

Has this result been shown before? Perhaps a weaker result: if we take $t$ to be on a compact interval $[-\varepsilon, \varepsilon]$, can we guarantee that the injectivity radius of $N_t$ does not shrink to zero as $t\rightarrow \varepsilon$?

Any help would be greatly appreciated!