Davide Legacci | Research Blog

Tags: geometry

An exercise on submanifolds arising as regular level sets of smooth maps¹.

An exercise on regular level sets

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Let $F:{\mathbb{R}}^3\to {\mathbb{R}}^2$ be $F(x,y,z)=(x^2,y^2-z^2)$. Its differential $d{F}_{(x,y,z)}:R^3\to {\mathbb{R}}^2$ is the linear map

$$d{F}_{(x,y,z)}=\left(\begin{array}{ccc}2x& 0& 0\\ 0& 2y& -2z\end{array}\right).$$

A point $p\in {\mathbb{R}}^3$ is a critical point for $F$ if $d{F}_p$ is not surjective, i.e. if $\text{rk}(d{F}_p)<2$.

A point $(\alpha ,\beta )\in {\mathbb{R}}^2$ is a critical value for $F$ if its level set $F^{-1}(\alpha ,\beta )$ contains a critical point. By the regular level set theorem, if $(\alpha ,\beta )$ is not a critical value – i.e., its level set $F^{-1}(\alpha ,\beta )$ does not contain critical points – then $F^{-1}(\alpha ,\beta )$ is a submanifold of ${\mathbb{R}}^3$ of codimension $2$.

The critical points for $F$ are $(0,y,z)$ for all $y,z\in \mathbb{R}$; and $(x,0,0)$ for all $x\in \mathbb{R}$. Thus for all $\alpha >0$ and $\beta >0$, the value $(\alpha ,\beta )\in {\mathbb{R}}^2$ is regular and its level set $S=(x,y,z):F(x,y,z)=(\alpha ,\beta )$ is a non-empty $1$-dimensional submanifold of ${\mathbb{R}}^3$ given in parametric form by

$$\begin{array}{rl}{x}^2& =\alpha ,\\ y^2-z^2& =\beta .\end{array}$$

This is a disconnected manifold given by $4$ hyperbole branches:

The branch with $x>0$ and $y>0$ is parametrized by $\psi :\mathbb{R}\to {\mathbb{R}}^3$ with

$$\psi (t)=(x,y,z)(t)=(\sqrt{\alpha },\sqrt{\beta +t^2},t).$$

Differentiating the parametrization with respect to $t$ gives the vector field ${\partial }_t$ spanning the tangent space to $S$ at $t$ as a linear combination of the basis vector fields ${\partial }_x,{\partial }_y,{\partial }_z$: we get ${\partial }_t=(0,z/y,1)$, so the affine tangent space to $S$ at $(x,y,z)$ is

$${\tilde{T}}_{(x,y,z)}S=T_{(x,y,z)}S+(x,y,z)=⟨(0,z,y)⟩+(x,y,z).$$

For example for $t=1$, $\alpha =\beta =1$, $(x,y,z)=(1,\sqrt{2},1)$, we get

$${\tilde{T}}_{(1,\sqrt{2},1)}S=⟨(0,1,\sqrt{2})⟩+(1,\sqrt{2},1).$$

1. J. M. Lee, Introduction to Smooth Manifolds, 2nd ed. in Graduate Texts in Mathematics. Springer-Verlag New York, 2012. ↩