EDB — 1H3

[1H3] Prerequisites: [0JV]↺↻, [0JY]↺↻, [0T5]↺↻, [0T7]↺↻, [1DT]↺↻, [1GD]↺↻, [1P1]↺↻ and [1PG]↺↻.
Difficulty:**.

For this exercise we need definitions and results presented in the Chapter [1NT]↺↻.

Let $r\ge 1$ integer, or $r=\infty$. Let $F:{ℝ}^2\to ℝ$ of class $C^r$, and such that $\nabla F\ne 0$ at every point $F=0$.

We know, from [0JV]↺↻, that $\{F=0\}$ is the disjoint union of connected components, and from [0JY]↺↻ that every connected component is a closed.

Show that, for every connected component $K$, there is an open set $A\supseteq K$ such that $K=A\cap \{F=0\}$, and that therefore there are at most countably many connected components.

Show that each connected component is the support of a simple immersed curve of class $C^r$, of one of the following two types:

• the curve is closed, or

• The curve $𝛾:ℝ\to {ℝ}^2$ is not closed and is unbounded (i.e. $\underset{t\to \pm \infty }{lim}|𝛾(t)|=\infty$).

The first case occurs if and only if the connected component is a compact.