Exercise 5, § 2

Soit $r$ an element of $\mathbf{N}\cup \{\infty , \omega \}$, let $n$ and $d$ be integers, and let X be a submanifold of class $C^r$ of $\mathbf{R}^n$, closed, purely of dimension $d$.[^1]

a) Prove that there exists a neighbourhood U of X in $\mathbf{R}^n$ and an isomorphism of class $C^r$ of $X\times ]-1; 1[^{n-d}$ onto U.

b) Every path in X is strictly homotopic to a path of class $C^r$. c) If two paths of class $C^r$ in X are homotopic, they are connected by a homotopy of class $C^r$.

[^1]: According to theorems of Whitney and Grauert, every variety of class $C^r$, of finite dimension and whose topology is of countable type is isomorphic to a closed submanifold of a numerical space.