# Injectivity radius of parallel hypersurfaces

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!

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 $$|t| will correspond to the slice $$\{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 $$|t|, we have that the injectivity radius from $$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 $$t \to 1$$.

• 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 $N$. – Ryan Vaughn Apr 2 '20 at 17:21
• Thank you for the great edit! This was exactly what I was looking for. – Ryan Vaughn Apr 3 '20 at 13:46