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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!

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Yes, the two injectivity radii are related.

Let $r>0$ be the injectivity radius from $N$. Then there is a neighborhood $U \subset M$ of $N$ and a Riemannian isometry $\phi : U \to (-r,r) \times N$. Each hypersurface $N_t$ as you describe, for $|t|<r$ will correspond to the slice $\{t\} \times N$ with the identification provided by $\phi$.

Identifying $U$ with $(-r,r)\times N$, the Riemannian metric has the form $dx^2 + h_x$ where, for all $x\in (-r,r)$, $h_x$ denotes a smooth one-parameter family of Riemannian metrics on $N$. This fact is just a generalization of the concept of normal coordinates (this is for sure proved in Gray's book, I think).

Furthermore, for all $p\in N$, the rays $\tau \mapsto (\tau,p)$, for $\tau \in (-r,r)$ are unit speed geodesics that realize the distance to $N$, that is $d_N((\tau,p)) = |\tau|$.

Therefore, if $|t|<r$, we have that the injectivity radius from $N_t$ is equal to $r-|t|$.


EDIT AFTER COMMENT

After Ryan's comment, I realize what the OP was really asking concerns the injectivity radius of the embedded submanifold $N_t$ with the metric induced by $(M,g)$, that is the injectivity radius of $(N,h_t)$, where $h_t$ is a one-parameter family of metrics on a compact manifold $N$ (induced by the ambient metric $g$, in this case, but this is not really important).

Since the function $h \mapsto i_h(N)$, defined on the space of smooth Riemannian structure equipped with the $C^2$ topology, is continous, and since the aforementioned family $h_t$ is smooth in $t$, for small $t$, the result easily follows. The continuity of $h \mapsto i_h(N)$ is proved here:

Ehrlich, Paul E., Continuity properties of the injectivity radius function, Compos. Math. 29, 151-178 (1974). ZBL0289.53034.

and see also the following reference for perhaps simpler statements and a more accessible proof:

Sakai, Takashi, On continuity of injectivity radius function, Math. J. Okayama Univ. 25, 91-97 (1983). ZBL0525.53053.

Since the proof above is not simple, let me show how one can prove your weaker claim. For a given manifold $(N,h)$ there exist a beautiful universal lower bound for its injectivity radius given explicitly in terms of a lower bound $\delta$ for the Ricci curvature, an upper bound $\Delta$ for the sectional curvature, the diameter $D$ and the volume $V$. This is originally due to Cheeger, but the lower bound can be recovered by more "elementary" arguments such as (i) the Klingenberg Lemma and (ii) the Heintze-Karcher inequality (cf. Note 6.5.2.2 and references therein in Berger's "Panoramic view of Riemannian geometry). In particular one obtains

$$ i_h(N) \geq \inf\left\{ \frac{\pi}{\sqrt{\Delta}}, c_n \left(\frac{\sqrt{|\delta|}}{\sinh(\sqrt{|\delta|} D)}\right)^{n-1} V \right\} $$

where $c_n$ is a universal dimensional consant.

Applyting this estimate to your specific case $(N,h_t)$ shows that the injectivity radious cannot become zero for small $t$.

Clearly $t \mapsto i_{h_t}(N)$ can become zero when your family $N_t$ approaches the normal cut locus, as $N_t$ can collapse. Consider for example the unit circle in $\mathbb{R}^2$, for which $N_t$ collapses to a point as $t \to 1$.

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  • $\begingroup$ Interesting. I don't quite follow the last sentence. It seems you are talking about the "normal cut locus" to $N$, that is, the distance $r$ for which the geodesics intersecting $N$ orthogonally are unique and length minimizing. I am interested in the injectivity radius of the submanifold $N_t$, i.e. the relationship between the injectivity radius of $N$ endowed with the metric $h_x$ as compared to the metric on $N$. $\endgroup$ – Ryan Vaughn Apr 2 at 17:21
  • $\begingroup$ Thank you for the great edit! This was exactly what I was looking for. $\endgroup$ – Ryan Vaughn Apr 3 at 13:46

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