What does the Jacobian matrix of the projection mapping for Normal bundle look like? (2.3.14 G&P)

Question

I want to solve this question:

I feel like the previous question is similar to the one given in this link:

Natural projection of tangent bundle is submersion

Am I correct? but what does the Jacobian matrix look like in our situation here?

Thanks.

Answer 1

I don't recall all the notation from G&P so I'll try to just explain why the projection from the normal bundle of a manifold $M$ to the manifold is a submersion. Let's write this as $\pi:NM\to M$. A trivializing open neighborhood for $NM\to M$ is an open neighborhood $U\subseteq M$ so that $\pi^{-1}(U)\cong U\times \mathbb{R}^k$. If $U$ is a trivializing open neighborhood for $NM$ with $U$ also a coordinate chart neighborhood on $M$ with coordinates $(x^1,\ldots, x^m)$, then on $\pi^{-1}(U)$ the bundle is parametrized by $(x^1,\ldots, x^m, v^1,\ldots, v^k)$.

If we think about the projection $\pi^{-1}(U)\to U$ and put it in coordinates, we are just forgetting about the last $k$ data above, so the map is locally $(x^1,\ldots, x^m, v^1,\ldots, v^k)\to (x^1,\ldots, x^m).$ The associated Jacobian matrix is of the form $J=[I_m|0_k]$ where $I_m$ is the $m\times m$ identity matrix and $0_k$ is the $k\times k$ zero matrix. Notice that $J$ is a surjective linear map, so that $\pi: NM\to M$ is a submersion.