Given an $m$-dimensional manifold $M$, and an $n$-dimensional submanifold $N$, with $n<m$, the tangent bundle of $N$ is a smooth $n$-dimensional subbundle of $TM|_N$.
When can $TN$ be extended to an $n$-dimensional subbundle of $TM$?
There are obvious examples where it's easy to extend $TN$ to a subbundle of $TM$, but for a spiral in $\mathbb{R}^2$, or the equator in a sphere it is not possible.
Edit - It's interesting to note that a sphere doesn't admit any $1$-dimensional subbundle. On the other hand, $\mathbb{R}^2$ obviously admits a $1$-dimensional subbundle, but still the tangent bundle of the spiral can't be extended to a full subbundle. Similar "spiral"-like submanifolds of any dimension $1 \le n < m$ can be found for every $m$-dimensional manifold.
Cross posted to MO