I am having much trouble reading the proof Local Immersion Theorem in Guillemin & Pallock's Differential Topology on Page 15.
Now we try to augment $g$ so that the Inverse Function Theorem may be applied.
How does it mean here to "augement $g$"?
As $dg_0: \mathbb{R}^{k} \rightarrow \mathbb{R}^{l}$ is injective.
Why $dg_0$ is injective?
Could some one explain these for me? Thanks.
The Inverse Function Theorem requires that you are mapping between spaces of the same dimension. In the present case, you are mapping from a space of dimension $k$ to a space of dimension $l$, where $l \geq k$. Hence we need to modify $g$ somehow to get a map between spaces of, say, dimension $l$. Guillemin and Pollack achieve this by using $U \times \Bbb R^{l-k}$ in place of $U$ and making a new function $G: U \times \Bbb R^{l-k}$ that acts as $g$ on the $U$ part and as the identity on the $\Bbb R^{l-k}$ part.
The way the proof is set up, we have that $$g = \psi^{-1} \circ f \circ \phi,$$ where $f$ is an immersion at $x \in X$ and $\phi$, $\psi$ are local parametrizations. This means that $df_x$ is injective and $d\phi_0$, $d(\psi^{-1})_0$ are isomorphisms. By the chain rule, $$dg_0 = d(\psi^{-1})_0 \circ df_x \circ d\phi_0,$$ from which it follows that $dg_0$ is injective.
