# Tangent spaces of two transverse subspaces are transverse subspaces

I am very new to differential geometry and was thrown this very long question:

Suppose that two subspaces $$V$$ and $$W$$ of $$\mathbb{R}^n$$ are transverse (so $$\text{Span}(V,W)=\mathbb{R}^n$$). Let $$O$$ be an open neighborhood of $$0\in \mathbb{R}^n$$ and define $$V':=V\cap O$$, $$W':=W\cap O$$. Suppose $$\phi:O\rightarrow U$$ is a diffeomorphism from O to an open neighborhood $$U$$ of $$0\in \mathbb{R}^n$$ with $$\phi(0)=0$$.

• Prove that $$\phi(V')$$ and $$\phi(W')$$ are sub-manifolds of $$\mathbb{R}^n$$

• Prove that $$T_0\phi(V')$$ and $$T_0\phi(W')$$ are transverse subspaces of $$\mathbb{R}^n$$

• Hence show there is no diffeomorphism from an open neighborhood of $$(0,0)$$ to another open neightborhood of $$(0,0)$$ which fixes the x-axis but takes the y-axis to the graph of $$y=x^3$$

Here is what I've got so far: To show that we have submanifolds: The question gives us the fact that there is an open neighborhood $$O$$ of $$0$$ and an open set $$U$$ containing $$0$$ with a diffeomorphism $$\phi:O\rightarrow U$$ such that $$\phi(0)=0$$. To show that $$\phi(V')$$ is a k-dimensional embedded submanifold of $$\mathbb{R}^n$$, we also want to show that $$\phi(O\cap \mathbb{R}^k)=U\cap \phi(V')$$. Note that $$U\cap \phi(V')=U\cap\phi(V\cap O)=U\cap\phi(V)\cap\phi(O)$$ since $$\phi$$ is a bijection. Furthermore, this is just $$U\cap\phi(V)$$ since $$\phi(O)=U$$. So we need to see that $$\phi(O\cap \mathbb{R}^k)=U\cap \phi(V)$$ Here's where I'm stuck on this part. If this were just $$\phi(O)$$, we could say that clearly $$U\subseteq \phi(V)$$ since $$\phi$$ will take open sets around the origin to other open sets around the origin. But I'm not really sure how this goes seeing as $$U$$ and $$\phi(V)$$ might not even be $$k$$-dimensional.

Next, for the transverse tangent spaces: I know that $$T_0\phi(V')\cap T_0\phi(W')=T_0(V'\cap W')$$ so we can show they intersect in a $$k$$-dimensional subspace just like $$V'$$ and $$W'$$ do, but I don't know how to get to transversality from that.

For the last part, I'm assuming it will be more obvious to me once I have a good grasp of the rest. Any help is appreciated!

• Do you know that diffeomorphisms map submanifolds to submanifolds? Can you show that in general? Next, if $L$ is an invertible linear map on $\Bbb R^n$ and $V=\text{Span}(v_1,\dots,v_k)$, then $L(V) = \text{Span}(Lv_1,\dots,Lv_k)$? – Ted Shifrin Feb 11 at 17:35