(a) State and prove the inverse function theorem for a smooth map $f: X \rightarrow Y$ between manifolds without boundary.

[You may assume the inverse function theorem for functions in Euclidean space.]

(b) Let $p$ be a real polynomial in $k$ variables such that for some integer $m \geqslant 1$,

$$p\left(t x_{1}, \ldots, t x_{k}\right)=t^{m} p\left(x_{1}, \ldots, x_{k}\right)$$

for all real $t>0$ and all $y=\left(x_{1}, \ldots, x_{k}\right) \in \mathbb{R}^{k}$. Prove that the set $X_{a}$ of points $y$ where $p(y)=a$ is a $(k-1)$-dimensional submanifold of $\mathbb{R}^{k}$, provided it is not empty and $a \neq 0$.

[You may use the pre-image theorem provided that it is clearly stated.]

(c) Show that the manifolds $X_{a}$ with $a>0$ are all diffeomorphic. Is $X_{a}$ with $a>0$ necessarily diffeomorphic to $X_{b}$ with $b<0$ ?