Lower bound for the normal injectivity radius

Question

Let $(M,g)$ be a closed Riemannian manifold and let $N$ be a closed embedded submanifold. A tube $T(N,r)$ of radius $r$ of $N$ is defined as the set of points of $M$ which can be reached by a geodesic path of length $< r$ starting from $N$ orthogonally. We further require that $T(N,r)$ is open in $M$ and that the map $(p,v)\mapsto \exp_{p}(v)$ defined on $\{(p,v)\in TN^{\perp}\,\,:\,\, g_p(v,v)<r^2\}$ is a diffeomorphism onto $T(N,r)$. Then the normal injectivity radius of $N$ (which we denote by ${\rm nir}(N)$) is defined as the maximal radius of a tube around $N$.

Assume that the sectional curvatures of $M$ and the second fundamental form of $N$ are uniformly bounded by a constant $K$. Is it possible to show that ${\rm nir}(N)\geq c(K)$, for some positive constant $c(K)$ depending only on $K$? Can this bound be obtained explicitly?

Answer 1

This is false as can be seen already on curves in the plane. Consider a "Dumbbell" shaped curve in the plane.

Assume that the two "weights" are roughly circles of radius 1. If these two circles are far enough, they can be joined smoothly by a neck as thin as you like in the middle while the curvature of the curve stays less than 1.

The normal injectivity radius of these examples are as small as one wants.

Answer 2

This is false also if you add some conditions on $M$. It seems that you must add more dependency on $N$ in order to get a lower bound.

Fix a natural $n$ and consider the subtorus $N_n$ in $M=\mathbb{R}^2/\mathbb{Z}^2$ given as the image of $[0,1]$ under $x\mapsto (x,nx)$. Here $M$ is a fixed flat Riemannian manifold of volume 1 and $N_n$ is a sequence of flat submanifolds of arbitrarily small nir. You can also have such examples in higher dimensions, of course (and also in $\dim(M)=1$; I hesitated to give a trivial counterexample...).