Let $\alpha \colon I \to \mathbb{R}^{n+1}$ be a parametrized curve with $\alpha(t) \in S_1 \cap S_2$ for all $t \in I$, where $S_1$ and $S_2$ are two $n$-surfaces in $\mathbb{R}^{n+1}$.
Suppose that $\mathbf{X}$ is a vector field along $\alpha$ which is tangent to both $S_1$ and to $S_2$ along $\alpha$.(a) Show by example that $\mathbf{X}$ may be parallel along $\alpha$ viewed as a curve in $S_1$ but not parallel along $\alpha$ as a curive in $S_2$.
(Original image here.)
I stuck in above question. I have some idea but don't know how to proceed.
Idea: Consider a sphere of unit radius as one surface and remove one point from $X – (0,0,-1/2)$ and this acts as the other surface. Now define a path as $(0.75 \cos t,0.75 \sin t, -1/2)$ and then define a vector field as $\mathbf{X} = (0.75 \cos t, 0.75 \sin t, 0)$. These satisfy all conditions but I cannot reach the final result.
Any other example is also welcome but with more of geometric intution.
Best Answer
Take a sphere and intersect it with a plane going through the origin. The intersection is a geodesic of the sphere, so there is a nice candidate for the vector field. Once you figure out the vector field, it should be easy to show that it is not parallel when you view it on the plane.