My motivation is the recent question I just answered, and my answer use too many hypothesis that I considered superfluous: Always "one double root" between "no root" and "at least one root" ? (Second version)
So now on to the question:
Consider a differentiable $n$-dimensional manifold $M$.
Let $f:\Bbb S^{k}\rightarrow M$ be a differentiable bijective embedding of the $k$-sphere ($k\leq n$) into $M$.
Let $H:[0,1]\times \Bbb > S^{l}\rightarrow M$ be a differentiable function such that $H(t,\cdot)$ is bijective for all $t$ (this is a homotopy of an embedding of a $l$-sphere in $M$).
Further, $\operatorname{range}(H(0,\cdot))$ intersects $\operatorname{range}(f)$ but $\operatorname{range}(H(1,\cdot))$ does not intersect $\operatorname{range}(f)$.
The question is: must there exist a $t$ and a point $p\in M$ such that between the tangent space of $\operatorname{range}(H(t,\cdot))$ at $p$ and the tangent space of $\operatorname{range}(f)$ at $p$, one must contains the other?
I am sorry if this question have been answered before, or is even a standard theorem, or if this question is poorly phrased. I barely studied differential geometry. Thank you for your help.
EDIT: Why the title?
Consider a floating blob of water, and we have a thread submerged in it. Well, for simplicity, the question simply assume that the thread is a loop (which is essentially equivalent to a thread fixed at infinity) and we assume that we are already part way through pulling the thread out, so that part of the thread is the on the surface of the water blob. The surface of the water blob is the image of $f$, and the thread at time $0$ is the image of $H(0,\cdot)$. Now imagine we pull the thread out, it seems like at the last moment where the thread touch the water, at the point it touches, the tangent line to the thread is also tangent to the water surface at that point. Hence the question's title.