Lower bound of the injectivity radius

Question

Suppose $F:M^n \rightarrow \mathbb{R}^m$ is an isometric immersion with bounded second fundamental form. This means the following: There exists a constant $C>0$ such that for any $p \in M^n$ and for any orthonormal basis $e_1,\ldots,e_n$ of $T_pM$ we have $$ |h(e_i,e_j)| \leq C$$ where $h$ is the second fundamental form defined by $$ h(X,Y) = \text{normal part of } D_{dF(X)} dF(Y) $$ for vector fields $X$ and $Y$ on $M$ and where $D$ is the regular derivative on $\mathbb{R}^m$.

Does there exist a lower bound for the injectivity radius on $M$ (note we do not assume $M$ to be compact)? The injectivity radius is defined as the lowest radius for which the exponential map is a diffeomorphism.

Answer 1

The Gauss equation $$R(X,Y,Z,W) = h(X,Z)h(Y,W) - h(X,W)h(Y,Z)$$ tells us that for any orthonormal pair $(e_1,e_2)$ we have $$\operatorname{sec}(e_1,e_2) = h(e_1,e_1)h(e_2,e_2)-h(e_1,e_2)^2;$$ so your bound on the second fundamental form implies the sectional curvature of $M$ is bounded by $C^2.$ Assuming $M$ is complete, we can now apply the usual injectivity radius estimate from intrinsic Riemannian geometry, which tells us that $$\operatorname{inj}(M) \ge \min \left\{ \frac{\pi}C, \frac12 \ell(M) \right\}$$ where $\ell(M)$ is the infimum of lengths of closed geodesics in $M$. (See e.g. Section 6.6 of Petersen's Riemannian Geometry 2ed.)

It remains to control $\ell(M).$ This is easy: for any unit-speed closed geodesic $\gamma$ in $M$, we know that the extrinsic curvature of $\gamma$ in $\mathbb R^m$ is given by $\kappa_\gamma = |h(\dot \gamma, \dot \gamma)| \le C,$ so we must have $$\ell(\gamma)\ge\frac{2\pi}C\;\text{and thus}\;\ell(M)\ge\frac{2\pi}C.$$ Thus we conclude $$\operatorname{inj}(M) \ge \frac{\pi}C.$$