Consider a smooth compact manifold $M$ of dimension $n$, with or without boundary. Choose a submanifold $N$ of $M$ of dimension $k$, where $1 \leq k \leq n - 1$, such that $N$ is either without boundary, or $\partial N \subset \partial M$. My question is, are there well-known necessary/sufficient conditions which dictate whether $N$ can be extended to a $k + 1$ dimensional submanifold $N'$ such that $N \subset N'$, and $N'$ is either without boundary, or $\partial N' \subset \partial M$?

Note: Heavily edited after Ryan Budney's comment (the initial question had incorrect notation). Budney's comment also shows that an extension is not always possible. But I would like to understand some conditions which guarantee or disallow the existence of such extensions.

1more comments