In the course of my research, I'm trying to prove properties about geodesics using the canonical Euclidean metric for a certain class of manifolds called "aligned manifolds". First, I'll describe aligned manifolds and then state the main question that I have.
Aligned Manifolds
First, let $E$ be an $n$-dimensional Euclidean coordinate system. Let $v = (v^1,v^2,...,v^n)$ be an $n$-dimensional vector in $E$. We define the sign of $v$ as:
$$sign(v) = (sign(v^1),sign(v^2),...,sign(v^n))$$
where the $sign$ of a scalar is 1, 0, or -1 if the scalar is positive, zero, or negative respectively.
A connected manifold $M$ is aligned with respect to $E$ if:
$$\forall x,y\in M\ldotp \forall v_x\in T_xM\ldotp \exists x_y\in T_yM\ldotp sign(v_x)=sign(v_y)$$
In other words, for every pair of points $x$ and $y$ in $M$ and every vector in the tangent space at $x$, there exists a vector in the tangent space at $y$ that points in the same overall direction. In 2 dimension, both tangent vectors point to the same quadrant; in 3 dimesions, both point to the same octant, etc. One can define aligned manifolds using the co-tangent space instead of the tangent space to describe the same set of manifolds.
Examples:
In $\mathbb{R}^2$: The open arc of a circle from 0 degrees to $\frac{\pi}{2}$ degrees is aligned.
The arc of a circle from $\frac{\pi}{4}$ degrees to $\frac{3\pi}{4}$ degrees is not aligned.
In $\mathbb{R}^3$: A manifold defined as $\{x^2+y^2=1,x>0,y<0,z>x+y^2\}$ is aligned.
A hemi-sphere of a 2-sphere is not aligned.
Main Question
I am trying to prove the following proposition:
Proposition. Let $M$ be an aligned manifold. For any pair of points $x,y\in M$, there exists at most one geodesic $\gamma: \mathbb{R}\rightarrow M$ such that $\gamma(0)=x$, $\gamma(1)=y$, and $\forall t\in[0,1]\ldotp \gamma(t)\in M$.
This is equivalent to asking if there do not exist any conjugate points in $M$. I've been trying to tackle this problem by showing that a Jacobi field which vanishes in $M$ cannot exist, although I haven't had much luck. The Cartan theorem gives us geodesic completeness if sectional curvature is non-positive; however, aligned manifolds may have positive curvature.
I would greatly appreciate any suggestions of how to go about proving this proposition or alternatively, a counter-example that demonstrates that the proposition is false.