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I'm new in the subject of manifolds, and I do not know very much about them: I just think about it as subsets of some n-dimensional Eculidean space, that are smooth etc. But I don't know/cannot use the rigorous definitions of manifolds at the moment. So please can somebody help me :)

Let $$ G=\{(x,y,z)\in \mathbb{R^3|} x^2+y^2-z^2+1=0; z>0\} $$

Question 1: Is $h(u,v) = (u,v, \sqrt{u^2+v^2+1}) $ a correct parameterization?

Question 2: How can I show that this is a 2-dimensional submanifold of $\mathbb{R}^2$ ?

I've got two equivalent definitions of manifolds, namely:

Defintion 1 A nonempty subset M of $\mathbb{R}^n$ is a $k$-dimensional $C^l$-submanifold of $\mathbb{R}^n$, $l\ge1$, if for each $p\in M $ there is a neighborhood $U$ in $\mathbb{R}^n$ and a $C^l$-map $F:U \rightarrow \mathbb{R}^{n-k} $ such that

  1. rank$D_xF=n-k$, for $x\in M\cap U$
  2. $M\cap U = \{x\in U|F(x)=0 \}$

Definition 2 A nonempty subset M of $\mathbb{R}^n$ is a $k$-dimensional $C^l$-submanifold of $\mathbb{R}^n$, $l\ge1$, if for each $p\in M $ there is a neighborhood $U$ in $\mathbb{R}^n$ and a $C^l$-map $h:V \rightarrow U\cap M$ of an open set $V \subset \mathbb{R}^k $ onto $U\cap M \subset \mathbb{R}^n$ such that

  1. $h$ is differentiable
  2. $h$ is a homeomorphism
  3. For each $q\in V$ the derivative $D_qh: \mathbb{R}^k \rightarrow \mathbb{R}^n $ is one-to-one

Which one should I use? How?

Should I just show that the rank of $D_xF$ equals 1? Where $F(x,y,z)=x^2+y^2-z^2+1=0$? Or should I show that my parameterization is differentiable, a homeomorphism and for each point that the derivative $D_qh$ is a bijection?

Applied mathematician
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1 Answers1

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In this case the easiest way to go is to use the so called submersion property : if $f : \mathbb{R}^n \rightarrow \mathbb{R}^p$ is a smooth function such that $D_xf$ is surjective on an open neighborhood of $M=f^{-1}(0)$ then $M$ is a submanifold of $\mathbb{R}^n$ of dimension $n-p$.

Here $f(x,y,z)=x^2+y^2-z^2+1$, and you just need to check that the differential does not vanish near the set $M$ of its zeroes. This will prove that $M$ is a 2-dimensional submanifold of $\mathbb{R}^3$ (i.e. a surface).

Your parametrization is correct.

Albert
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