1
$\begingroup$

Let $M$ be a closed smooth $m$-manifold, $I\subset\mathbb{R}$ be an interval, $(F_t:M\to\mathbb{R}^n)_{t\in I}$ be a smooth family of embeddings $M_t:=F_t(M)$ and $A_t$ be the 2nd fundamental form of $F_t$. We equip the Euclidean metric on $\mathbb{R}^n$. Put $\pi_t:\mathcal{N}_t\to M$ be the normal of the embedding $F_t$, let

\begin{align} \Psi_t:\mathcal{N}_t\ni v\to \pi_t(v)+v\in\mathbb{R}^n \end{align} and $\mathcal{N}_{t;\delta}:=\{v\in\mathcal{N}_t\ |\ |v|<\delta\}\quad (\delta>0)$.

Q.1: Suppose that $|A_t|^2$ is bounded uniformly in $t\in I$. Then does it hold that $\exists \delta_0>0$ s.t. $\forall t\in I$ the map $\Psi_t$ is a diffeomorphism from the set $\mathcal{N}_{t;\delta_0}$ to its image?

Q.2: Suppose that the interval $I$ equals to $[0,1]$. Then does the same assertion as in Q.1 hold?

Thank you.

  • 1
    Are you requiring the interval $I$ to be closed? If so I think this is true by some compactness argument without needing the bound on $A$. If you allow an open interval then this is false - you should be able to construct an embedding with bounded curvature that comes arbitrarily close to itself. – 2017-02-06

0 Answers 0