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 \(r≥ 1\) integer, or \(r=∞\). Let \(F:ℝ^ 2→ℝ\) of class \(C^ r\), and such that \(∇ F≠ 0\) 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 \(A⊇ K\) such that \(K=A∩ \{ F=0\} \), 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 \(𝛾:ℝ→ℝ^ 2\) is not closed and is unbounded (i.e. \(\lim _{t→ ±∞}|𝛾(t)|=∞\)).

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