Almost geodesic on non complete manifolds

Question

Let $M$ be a connected manifold equipped with a connection $\nabla$. By Hopf-Rinow theorem, we know that if $M$ is complete then for any $x,y$ there exist a curve $\gamma:[0,1] \to M$ such that $\gamma(0) = x, \gamma(1) = y$ and $\nabla_{\gamma'(t)} \gamma'(t)=0$ for all $t$. This is a way to say that $\gamma$ is a geodesic.

Suppose now that $M$ is possibly non complete. Given a threeshold $\varepsilon$, is it always possible to find a $\gamma$ between two fixed points $x,y$ such that $$ \frac{\lVert \nabla_{\gamma'(t)} \gamma'(t) \rVert }{\lVert \gamma' \rVert^2 } < \varepsilon $$

I kind of solved the case in which $M$ is of the form $\mathbb{R}^n \setminus \cup_{i=1}^k N_i $, where $N_i$ are submanifolds of codimension at least 2. In this case you can take a segment from $x,y$ and perturb it to be transverse to each $N_i$ in the $C_2$ topology (see Hirsch, differential topology, transversality chapter), so that $\gamma''$ will be almost zero and $\gamma'$ almost costantly $(y-x)$. Since $\dim \gamma + \dim N_i < \dim M = n$, transversality means $\textrm{Im} \gamma \subset M$. In my case this is enough to conclude, so this is just a curiosity :) Maybe something in the spirit of calculus of variations?

Answer 1

Start with the plane $\mathbb R^2$ and remove a slab, but keep a line going through the slab:

$$ Slab = \{(x, y) \in \mathbb R^2 : 0 < y < 1, x \neq 0\} $$ $$ M = \mathbb R^2 - Slab$$

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Note that $M_1$ is connected but curves going from one side of the slab to the other have a fixed direction for some time.

Now cut away a line-with-a-hole:

$$ Line_\delta =\{(x, y) \in \mathbb R^2 : y = 1 + \delta, x \neq 50\}$$ $$ N_\delta = \mathbb R^2 - Slab - Line_\delta $$

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At the top of the slab (the point $(1, 0)$) we will always have $\gamma'/|\gamma| = (0, 1)$. If $\delta$ is small enough compared to your $\epsilon$, you shouldn't be able to turn fast enough to avoid crashing into $Line_\delta$.

Edit: Another answer is to take the manifold

$$ ThickenedCircle_{r, \delta} = \{ p \in \mathbb R^2 : r-\delta < |p| < r+\delta \}.$$ First chose a sufficiently small $r$ so that the circle of radius $r$ does not obey your condition on the curve for $\epsilon/2$. Then if you chose $\delta$ small enough you get a flat incomplete 2-manifold where geodesics still must accelerate too much.