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!

  • 1
    $\begingroup$ 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 $L(V) = \text{Span}(Lv_1,\dots,Lv_k)$? $\endgroup$ – Ted Shifrin Feb 11 at 17:35

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