Interpreting the sum as a diffeomorphism

Question

Consider the manifold $M = \{(x,x^2) : x \in \mathbb{R}\}$ and for $\varepsilon > 0$ let $M_{\varepsilon} = \bigcup_{p \in M} B_{\varepsilon}(p)$. How can I find $\varepsilon > 0$ such that the map $$ F \colon NM \cap (\mathbb{R}^2 \times B_{\varepsilon}(0)) \rightarrow M_{\varepsilon} \colon (p,w) \mapsto p + w $$ is a diffeomorphism? The definition of the normal bundle is $$ N M = \{ (p,w) \in \mathbb{R}^2 \times \mathbb{R}^2 : p \in M \text{ and } w \in (T_{p} \, M)^{\bot} \}. $$

Answer 1

To get a diffeomorphism, you'll actually need a bijective (in particular, injective) immersion. Consider first the question of immersivity. Parametrize the curve $\alpha$ by arclength and then your map is given by $f(s,t) = \alpha(s) + t N(s)$ (where $N$ is the unit normal). You can check (using the Frenet equations) that $df$ has rank $<2$ at $(s,t)$ only when $t=1/\kappa(s)$. So if $t<1/\kappa_{\text{max}}=1/2$, the map will be an immersion.

Now check explicitly that if $t<1/2$ the normal line segments of length $t$ will never intersect, so the map is one-to-one. (By symmetry, the worst case scenario will be points $(\pm u,u^2)$.) Intuitively, since the parabola spreads apart, rather than getting close to itself, the curvature is the controlling phenomenon.