We are working through old qualifying exams to study. There were two questions concerning normal bundles that have stumped us:
$1$. Let $f:\mathbb{R}^{n+1}\longrightarrow \mathbb{R}$ be smooth and have $0$ as a regular value. Let $M=f^{-1}(0)$.
(a) Show that $M$ has a non-vanishing normal field.
(b) Show that $M\times S^1$ is parallelizable.
$2$. Let $M$ be a submanifold of $N$, both without boundary. If the normal bundle of $M$ in $N$ is orientable and $M$ is nullhomotopic in $N$, show that $M$ is orientable.
More elementary answers are sought. But, any kind of help would be appreciated. Thanks.