Let M={$(x,y,z)\in \mathbb{R}^3$: $\frac{x^2} {a^2} + \frac {y^2}{b^2} + \frac{z^2}{c^2} =1$},where a,b,c>0.Prove that M is a smooth manifold.
I have proved M to be Hausdorff and 2nd countable but unable to prove that for each $p\in M$ there exists an open nbd. which is homeomorphic to open subset of $\mathbb{R}^n$.
To proving M is smooth, I have to take charts $(U_{\alpha}, \phi_{\alpha}$) and $(U_{\beta} , \phi_{\beta}$) such that $U_{\alpha} \cap U_{\beta}\neq \phi$ and I must show that $\phi_{\beta} \phi_{\alpha}^{-1} $ is $C^{\infty}$ function, which I have done.
So, Can you please tell how should I prove it locally homeomorphic to $\mathbb{R}^n$
Best Answer
Define a function $F:\mathbb R^3\to \mathbb R$ s.t. $(x,y,z)\mapsto x^2/a^2+y^2/b^2+z^2/c^2$, where $x,y,z$ are global coordinates of the real space (observe that you can define $M$ as the preimage $F^{-1}(1)$).
The function $F$ is smooth and it induces the differential map $dF_p:T_p\mathbb R^3\cong \mathbb R^3\to T_{F(p)}\mathbb R$, which is linear.
The representation of the differential is given, in this case, by the gradient of the function $$\nabla F\vert_p=(2x/a^2,2y/b^2,2z/c^2)^t$$ whose rank is less than $1$ iff $x,y,z=0$, but $\textbf 0\notin M\implies F$ is a submersion and $M=F^{-1}(1)$ is an embedded manifold of $\mathbb R^3$ of dimension $3-1=2.$