Today, the professor of Smooth Manifold's course proofed that $TS^{2n-1}$ is an embedded submanifold of $T\mathbb R^{2n}$ using the fact that $T\mathbb R^{2n}$ is diffeomorphic to $\mathbb R^{2n} \times \mathbb R^{2n}$ and describing $S^{2n-1}$ as inverse image of regular value. So, my question is:
If $M$ be a submanifold of $N$, is true that $TM$ is a submanifold of $TN$?
In above $TM$ denotes the tangent bundle of the smooth manifold $M$.
In short, yes. Pointwise, $T_pM\subseteq T_pN$, and indeed $M\subseteq N$. So at the very least we have that as sets $TM\subseteq TN$. You have to say a little bit more to show that it is a submanifold. Can you come up with coordinates for $TM$? Hint: use coordinates $(y^1,\ldots, y^m)$ in a small neighborhood $U$ of a point $p\in M$, and then use these to construct coordinates on $TM|_U$.
Actually, more is true. If $M\subseteq N$ is embedded, then $TM\subseteq TN$ is embedded also.
