Verify that a set is a submanifold

Question

I'm stuck on the following exercise:

"Verify that $M:=\{(x,y,z)\in\mathbb{R}^3:x^2+3y^2+2z^2=3, x+y+z=0\}$ is a submanifold. Also, say what its dimension is and compute the Jacobian matrix of the function."

The definition of $C^k (k\geq 1)$ submanifold of dimension $p$ that I have in my notes is:

"$V\subset\mathbb{R}^n$ is a $C^k (k\geq 1)$ submanifold of dimension $1\leq p\leq n-1$ iff $\forall x_0\in V \exists U$ neighbourhood of $x_0$ in $\mathbb{R}^n$ and a function $f\colon U\to\mathbb{R}^{n-p}$, $f\in C^k(U)$ such that: (1) $x_0\in U$, (2) $V\cap U=\{x\in U:f(x)=0\}$, (3)rank $Jf(x)=n-p\ \forall x\in U$."

It's the first exercise I do on (sub)manifolds and I don't know how to get started, so I would appreciate if someone explained to me how to do this kind of exercise.

Best regards,

lorenzo.

Answer 1

You have a set in $\mathbb{R}^3$ defined by 2 equations, so, as in linear algebra, you can start guessing that it will have dimension 1. If you are familiar with the equations, you will know that your are cutting a cone with a plane, which results in a curve.

Following your definition, I would define the function

$f(x,y,z)= ( x^2+3y^2+2z^2-3, x+y+z )$.

It is obvious that $f(p)=0$ if and only if $p\in M$. Now you have to check the other conditions.

Answer 2

You can use the fact that the pre-image of a regular value for a differentiable mapping $f: \mathbb{R}^n \to \mathbb{R}^m$ is a submanifold of dimension $n-m$. Here, our condition can be reduced to

$$3x^2 + 5y^2 + 4xy - 3 = 0, $$ so our $f: \mathbb{R^2} \to \mathbb{R}$ is given by $f(x,y) = 3x^2 + 5y^2 + 4xy - 3$. This reduces to showing that the Jacobian is full rank at your point (let me know if you're having problems with this).